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Week 8: Derivatives, Rules, Tangents & Critical Points

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Week 8: Derivatives, Rules, Tangents & Critical Points

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2026-09-12

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Mathematics I — Week 8: Derivatives, Rules, Tangents & Critical Points

IITM’s Week 8 covers differentiability and the derivative, computing derivatives, L’Hôpital’s rule, tangents, and critical points.

TL;DR

Week 7 built the limit machinery. Week 8 uses it to define the derivative:

\boxed{ f'(x)= \lim_{h\to0} \frac{f(x+h)-f(x)}{h} }

The derivative measures:

instantaneous rate of change
slope of the tangent line
local increase/decrease
where maxima/minima may occur

The highest-value ideas are:

derivative from first principles
differentiability vs continuity
power/product/quotient/chain rules
derivatives of exponential and logarithmic functions
tangent and normal lines
critical points
increasing/decreasing intervals
local maxima/minima
L’Hôpital’s rule for indeterminate forms

1. From secant slope to derivative

For two points on a curve:

(x,f(x))

and:

(x+h,f(x+h)),

the secant slope is:

\frac{f(x+h)-f(x)}{h}.

As:

h\to0,

the second point moves toward the first.

If the slope approaches a finite value, we define:

\boxed{ f'(x)= \lim_{h\to0} \frac{f(x+h)-f(x)}{h} }

This is the derivative.

2. Meaning of the derivative

The derivative can be interpreted in several equivalent ways.

Geometric meaning

f'(a)

is the slope of the tangent line at:

x=a.

Rate-of-change meaning

It tells how quickly:

f(x)

changes when

x

changes.

Local linear approximation

Near:

x=a,

the function behaves approximately like:

f(x)\approx f(a)+f'(a)(x-a).

That last view becomes increasingly important later.

3. Example from first principles

Take:

f(x)=x^2.

Then:

f'(x) = \lim_{h\to0} \frac{(x+h)^2-x^2}{h}.

Expand:

(x+h)^2=x^2+2xh+h^2.

Therefore:

f'(x) = \lim_{h\to0} \frac{2xh+h^2}{h}.

Factor:

= \lim_{h\to0}(2x+h).

Hence:

\boxed{ f'(x)=2x }

So at:

x=3,

the tangent slope is:

f'(3)=6.

4. Derivative notation

All of these may represent the derivative:

f'(x)

\frac{df}{dx}

\frac{dy}{dx}

D_xf

If:

y=f(x),

then:

\frac{dy}{dx}

means derivative of

y

with respect to

x

5. Differentiability

A function is differentiable at $x=a$ if:

f'(a)

exists as a finite real value.

That means:

\lim_{h\to0} \frac{f(a+h)-f(a)}{h}

exists.

6. Differentiability implies continuity

An important theorem:

\boxed{ \text{Differentiable at }a \Rightarrow \text{continuous at }a }

But the converse is false.

A function can be continuous but not differentiable.

7. Classic example: $|x|$

Consider:

f(x)=|x|.

It is continuous at:

x=0.

But the slope from the left is:

-1

and from the right:

+1.

So:

f'(0)

does not exist.

Therefore:

\boxed{ |x| \text{ is continuous but not differentiable at }0 }

8. Common reasons differentiability fails

A function may fail to be differentiable at:

a corner
a cusp
a vertical tangent
a discontinuity

So smoothness is stronger than continuity.

9. Constant rule

If:

f(x)=c,

then:

\boxed{ f'(x)=0 }

Why?

A horizontal line has slope:

0.

10. Power rule

For:

f(x)=x^n,

we have:

\boxed{ \frac{d}{dx}x^n = nx^{n-1} }

Example:

\frac{d}{dx}x^5 = 5x^4.

11. Examples

\frac{d}{dx}x^3=3x^2

\frac{d}{dx}x^2=2x

\frac{d}{dx}x=1

\frac{d}{dx}x^0=0

since:

x^0=1.

12. Constant multiple rule

If:

f(x)=c\,g(x),

then:

\boxed{ f'(x)=c\,g'(x) }

Example:

\frac{d}{dx}(7x^4) = 28x^3.

13. Sum rule

\boxed{ (f+g)'=f'+g' }

and:

\boxed{ (f-g)'=f'-g' }

Example:

f(x)=3x^4-2x^2+5x-7.

Then:

f'(x) = 12x^3-4x+5.

The constant:

-7

disappears.

14. Polynomial derivative

For:

P(x)=a_nx^n+\cdots+a_1x+a_0,

differentiate term by term:

\boxed{ P'(x) = na_nx^{n-1} +\cdots+a_1 }

The degree usually drops by:

1.

15. Product rule

For:

y=f(x)g(x),

you cannot simply multiply derivatives.

Instead:

\boxed{ (fg)'=f'g+fg' }

Mnemonic:

first derivative × second + first × second derivative.

16. Example: product rule

Let:

f(x)=x^2

and:

g(x)=x+1.

Then:

\frac{d}{dx}[x^2(x+1)]

equals:

2x(x+1)+x^2(1).

So:

=2x^2+2x+x^2

\boxed{ =3x^2+2x }

17. Quotient rule

For:

y=\frac{f(x)}{g(x)},

with:

g(x)\neq0,

we have:

\boxed{ \left(\frac fg\right)' = \frac{f'g-fg'}{g^2} }

18. Example: quotient rule

Take:

f(x)=\frac{x^2+1}{x}.

Let:

u=x^2+1, \qquad v=x.

Then:

u'=2x, \qquad v'=1.

Therefore:

f'(x) = \frac{2x(x)-(x^2+1)(1)}{x^2}.

So:

= \frac{2x^2-x^2-1}{x^2}

\boxed{ = \frac{x^2-1}{x^2} }

19. Chain rule

This is one of the most important rules in calculus.

If:

y=f(g(x)),

then:

\boxed{ \frac{dy}{dx} = f'(g(x))g'(x) }

Informally:

derivative of outside × derivative of inside.

20. Example: chain rule

Differentiate:

(3x+1)^5.

Outer function:

u^5.

Derivative:

5u^4.

Inner function:

u=3x+1.

Derivative:

3.

Therefore:

\boxed{ \frac{d}{dx}(3x+1)^5 = 15(3x+1)^4 }

21. Another chain-rule example

f(x)=(x^2+4)^3.

Derivative:

3(x^2+4)^2\cdot2x.

So:

\boxed{ f'(x)=6x(x^2+4)^2 }

22. Why the chain rule matters

Functions are often built by composition.

Week 5 gave:

(f\circ g)(x)=f(g(x)).

Week 8 asks:

How do derivatives behave under composition?

Answer:

\boxed{ (f\circ g)'(x) = f'(g(x))g'(x) }

So Week 5 composition now directly feeds into calculus.

23. Derivative of exponential

For the natural exponential:

\boxed{ \frac{d}{dx}e^x=e^x }

This is one reason

e

is mathematically special.

Its derivative is itself.

24. General exponential derivative

For:

a^x,

\boxed{ \frac{d}{dx}a^x = a^x\ln a }

Example:

\frac{d}{dx}2^x = 2^x\ln2.

25. Exponential with chain rule

For:

e^{g(x)},

\boxed{ \frac{d}{dx}e^{g(x)} = e^{g(x)}g'(x) }

Example:

\frac{d}{dx}e^{3x} = 3e^{3x}.

26. Derivative of logarithm

The fundamental logarithmic derivative is:

\boxed{ \frac{d}{dx}\ln x = \frac1x }

for:

x>0.

27. General logarithm

For:

\log_a x,

\boxed{ \frac{d}{dx}\log_a x = \frac1{x\ln a} }

28. Logarithm with chain rule

For:

\ln(g(x)),

\boxed{ \frac{d}{dx}\ln(g(x)) = \frac{g'(x)}{g(x)} }

where the logarithm is defined.

Example:

\frac{d}{dx}\ln(x^2+1) = \frac{2x}{x^2+1}.

29. Tangent line equation

Suppose we want the tangent to:

y=f(x)

at:

x=a.

The point is:

(a,f(a)).

The tangent slope is:

f'(a).

Use point-slope form:

\boxed{ y-f(a) = f'(a)(x-a) }

30. Example: tangent line

Find the tangent to:

f(x)=x^2

at:

x=2.

Point:

(2,4).

Derivative:

f'(x)=2x.

So:

f'(2)=4.

Tangent:

y-4=4(x-2).

Therefore:

\boxed{ y=4x-4 }

31. Normal line

The normal line is perpendicular to the tangent.

If tangent slope is:

m,

then normal slope is:

-\frac1m

when:

m\neq0.

So normal equation:

y-f(a) = -\frac1{f'(a)}(x-a).

32. Example: normal line

For:

f(x)=x^2

at:

x=2,

tangent slope:

4.

Normal slope:

-\frac14.

Thus:

\boxed{ y-4= -\frac14(x-2) }

33. Increasing functions and derivatives

If:

f'(x)>0

on an interval, then the function is increasing there.

Mental model:

positive tangent slope → graph moving upward.

34. Decreasing functions

If:

f'(x)<0,

then the function is decreasing.

Mental model:

negative tangent slope → graph moving downward.

35. Stationary point

A stationary point occurs when:

\boxed{ f'(x)=0 }

The tangent is horizontal there.

Possible interpretations:

local maximum
local minimum
neither

So:

f'(x)=0

does not automatically mean maximum/minimum.

36. Critical points

A critical point typically occurs where:

f'(x)=0

or where:

f'(x)

does not exist, provided the original function itself is defined there.

Critical points are candidates for extrema.

37. Example: quadratic critical point

Take:

f(x)=x^2-4x+1.

Then:

f'(x)=2x-4.

Solve:

2x-4=0.

Therefore:

x=2.

Then:

f(2)=4-8+1=-3.

So critical point:

(2,-3).

Since the parabola opens upward, this is a minimum.

38. First derivative test

Suppose:

f'(c)=0.

Inspect the sign of:

f'(x)

around

c

If:

+\to-

then function changes from increasing to decreasing.

So:

\boxed{\text{local maximum}}

If:

-\to+

then:

\boxed{\text{local minimum}}

If sign does not change

Then the critical point may be neither.

39. Example: first derivative test

Take:

f(x)=x^3-3x.

Derivative:

f'(x)=3x^2-3.

Factor:

f'(x)=3(x-1)(x+1).

Critical points:

x=-1,\quad1.

Check signs.

For:

x<-1,

f'(x)>0.

For:

-1<x<1,

f'(x)<0.

For:

x>1,

f'(x)>0.

Therefore:

At:

x=-1: \quad +\to-

local maximum.

At:

x=1: \quad -\to+

local minimum.

40. Second derivative

The derivative of the derivative is:

\boxed{ f''(x) }

called the second derivative.

It describes how the first derivative changes.

It is useful for concavity and classifying critical points.

41. Second derivative test

If:

f'(c)=0,

then:

If

f''(c)>0,

the graph is locally curved upward:

\boxed{\text{local minimum}}

If

f''(c)<0,

the graph is locally curved downward:

\boxed{\text{local maximum}}

If

f''(c)=0,

the test is inconclusive.

42. Example

Take:

f(x)=x^2-6x+5.

Then:

f'(x)=2x-6.

Critical point:

x=3.

Second derivative:

f''(x)=2.

Thus:

f''(3)=2>0.

Therefore:

x=3

is a local minimum.

This reproduces Week 3's quadratic vertex result using calculus.

43. Deriving the quadratic vertex with calculus

For:

f(x)=ax^2+bx+c,

differentiate:

f'(x)=2ax+b.

At the turning point:

f'(x)=0.

So:

2ax+b=0.

Thus:

\boxed{ x=-\frac b{2a} }

Exactly the Week 3 axis-of-symmetry formula.

So calculus explains why that formula works.

44. Critical point does not guarantee extremum

Consider:

f(x)=x^3.

Then:

f'(x)=3x^2.

So:

f'(0)=0.

But:

x=0

is neither local maximum nor minimum.

The function keeps increasing through it.

This is why you need a sign test or other classification.

45. Local vs global extrema

Local maximum

Higher than nearby values.

Global/absolute maximum

Higher than every value in the entire domain.

Similarly for minima.

A local extremum need not be global.

46. Optimization viewpoint

Many calculus problems can be phrased:

Find where some quantity is largest or smallest.

Typical workflow:


text
1. Build function
2. Differentiate
3. Solve f'(x)=0
4. Include domain boundaries if relevant
5. Evaluate candidates
6. Compare

This is one of the central applications of derivatives.

47. What is L’Hôpital’s rule?

Sometimes limits produce indeterminate forms like:

\frac00

or:

\frac{\infty}{\infty}.

Under appropriate conditions, L’Hôpital’s rule says:

\boxed{ \lim_{x\to a} \frac{f(x)}{g(x)} = \lim_{x\to a} \frac{f'(x)}{g'(x)} }

when the original ratio has one of the appropriate indeterminate forms and the derivative-limit exists suitably.

48. L’Hôpital example: $0/0$

Evaluate:

\lim_{x\to0} \frac{e^x-1}{x}.

Direct substitution:

\frac{0}{0}.

Differentiate numerator and denominator:

\frac{d}{dx}(e^x-1)=e^x

\frac{d}{dx}x=1.

Therefore:

\lim_{x\to0}\frac{e^x}{1} = 1.

So:

\boxed{1}

49. L’Hôpital example: $\infty/\infty$

Evaluate:

\lim_{x\to\infty}\frac{\ln x}{x}.

Both numerator and denominator grow:

\infty/\infty.

Apply L’Hôpital:

\lim_{x\to\infty} \frac{1/x}{1} = \lim_{x\to\infty}\frac1x = 0.

Thus:

\boxed{0}

This formally shows that:

x

grows faster than:

\ln x.

50. Another example

Evaluate:

\lim_{x\to\infty} \frac{x^2}{e^x}.

This is:

\frac{\infty}{\infty}.

First L’Hôpital:

\lim_{x\to\infty} \frac{2x}{e^x}.

Still:

\frac{\infty}{\infty}.

Apply again:

\lim_{x\to\infty} \frac{2}{e^x} = 0.

Hence:

\boxed{0}

This reflects the fact that exponentials eventually dominate polynomials.

51. L’Hôpital is not a quotient rule

Do not confuse:

Quotient derivative

\left(\frac fg\right)' = \frac{f'g-fg'}{g^2}

with L’Hôpital:

\lim\frac fg = \lim\frac{f'}{g'}.

L’Hôpital is a limit technique, not a differentiation formula.

52. When not to use L’Hôpital directly

You should not blindly apply it to:

\frac{5}{2}.

There is no indeterminate form.

Likewise:

\frac{0}{5}=0

is already determinate.

L’Hôpital is designed for specific indeterminate cases such as:

\frac00

and:

\frac{\infty}{\infty}.

53. Other indeterminate forms

You may also encounter:

0\cdot\infty

\infty-\infty

1^\infty

0^0

\infty^0.

Typically these first need algebraic/logarithmic rewriting before L’Hôpital can be applied.

For your immediate Week 8 prep, focus first on:

\boxed{ 0/0 \quad\text{and}\quad \infty/\infty }

54. Derivative sign table

Suppose critical points divide the line into intervals.

Create:

Interval	$f'(x)$	Behavior
first	$+$	increasing
second	$-$	decreasing
third	$+$	increasing

Then transitions tell you extrema.

This is often more reliable than trying to mentally visualize the entire function.

55. Example: analyze a polynomial

Take:

f(x)=x^3-3x^2.

Differentiate:

f'(x)=3x^2-6x.

Factor:

f'(x)=3x(x-2).

Critical points:

x=0,\quad2.

Now sign analysis:

$x<0$

Both

x

and

x-2

negative:

f'(x)>0.

Increasing.

$0<x<2$

x>0,\qquad x-2<0.

Thus:

f'(x)<0.

Decreasing.

$x>2$

Both positive:

f'(x)>0.

Increasing.

Therefore:

x=0

is local maximum.

And:

x=2

is local minimum.

56. Function values at those extrema

For:

f(x)=x^3-3x^2,

at:

x=0:

f(0)=0.

So local maximum:

(0,0).

At:

x=2:

f(2)=8-12=-4.

So local minimum:

(2,-4).

57. Tangent with horizontal slope

At a local max/min where differentiable:

f'(a)=0.

Therefore the tangent line is:

y=f(a).

A horizontal line.

This gives the geometric picture behind stationary extrema.

58. Derivative as units

Suppose:

s(t)

measures distance in kilometers and

t

is hours.

Then:

s'(t)

has units:

\frac{\text{km}}{\text{hour}}.

So the derivative is velocity.

If you differentiate again:

s''(t),

units become:

\frac{\text{km}}{\text{hour}^2}.

That corresponds to acceleration.

This is a good general mental model:

derivative = output units divided by input units.

Common mistake: derivative of a product

Wrong:

(fg)'=f'g'.

Correct:

\boxed{ (fg)'=f'g+fg' }

Common mistake: derivative of quotient

Wrong:

\left(\frac fg\right)' = \frac{f'}{g'}.

Correct:

\boxed{ \left(\frac fg\right)' = \frac{f'g-fg'}{g^2} }

The

f'/g'

expression belongs to L’Hôpital inside certain limits, not quotient differentiation.

Common mistake: chain rule omission

For:

(x^2+1)^5,

wrong:

5(x^2+1)^4.

You must multiply by derivative of the inside:

2x.

Correct:

\boxed{ 10x(x^2+1)^4 }

Common mistake:

f'(x)=0

means minimum

No.

It means only:

candidate stationary point.

You still need classification.

Possible:

maximum
minimum
neither

Common mistake: differentiability and continuity reversed

Correct:

\text{differentiable} \Rightarrow \text{continuous}.

Incorrect:

\text{continuous} \Rightarrow \text{differentiable}.

Counterexample:

|x|

at:

0.

Common mistake: using L’Hôpital unnecessarily

If:

\lim_{x\to2}\frac{x+1}{x+3},

direct substitution gives:

\frac35.

There is no need for L’Hôpital.

Prefer the simplest valid method.

65. Derivative formula sheet

Definition

\boxed{ f'(x)= \lim_{h\to0} \frac{f(x+h)-f(x)}h }

Constant

\boxed{ \frac d{dx}c=0 }

Power

\boxed{ \frac d{dx}x^n=nx^{n-1} }

Sum

\boxed{ (f+g)'=f'+g' }

Product

\boxed{ (fg)'=f'g+fg' }

Quotient

\boxed{ \left(\frac fg\right)' = \frac{f'g-fg'}{g^2} }

Chain

\boxed{ (f\circ g)'(x) = f'(g(x))g'(x) }

Natural exponential

\boxed{ (e^x)'=e^x }

Exponential

\boxed{ (a^x)'=a^x\ln a }

Natural logarithm

\boxed{ (\ln x)'=\frac1x }

66. Tangent/critical-point formula sheet

Tangent slope

\boxed{ m=f'(a) }

Tangent equation

\boxed{ y-f(a)=f'(a)(x-a) }

Critical point candidate

\boxed{ f'(x)=0 }

or derivative undefined while function is defined.

Increasing

\boxed{ f'(x)>0 }

Decreasing

\boxed{ f'(x)<0 }

67. L’Hôpital cheat sheet

For suitable differentiable functions:

\boxed{ \lim\frac{f(x)}{g(x)} = \lim\frac{f'(x)}{g'(x)} }

when the original form is:

\frac00

or:

\frac{\infty}{\infty}.

Workflow:


text
1. Substitute first
2. Confirm indeterminate form
3. Differentiate numerator
4. Differentiate denominator
5. Re-evaluate limit
6. Repeat only if still indeterminate

68. Week 7 → Week 8 connection

Week 7 gave:

\lim_{h\to0} \frac{f(x+h)-f(x)}h.

Week 8 names it:

f'(x).

Then:


text
limit
 ↓
derivative
 ↓
tangent slope
 ↓
increase/decrease
 ↓
critical points
 ↓
maxima/minima

This is the conceptual spine of introductory differential calculus.

69. What you should actually master

By the end of Week 8, you should be able to:

Explain derivative as instantaneous rate of change.
Compute a simple derivative from first principles.
Distinguish continuity from differentiability.
Use the power rule.
Differentiate polynomials.
Use product rule.
Use quotient rule.
Use chain rule.
Differentiate basic exponential functions.
Differentiate basic logarithmic functions.
Find tangent equations.
Find normal equations.
Solve: $f'(x)=0.$
Identify critical points.
Determine increasing/decreasing intervals.
Classify local maxima/minima with derivative signs.
Understand the second derivative test at a basic level.
Recognize $0/0$ and $\infty/\infty$ as indeterminate.
Apply basic L’Hôpital’s rule correctly.

If those are comfortable, Week 8 is under control.

Week 9 moves to integration of functions of one variable, computing areas, areas under curves, and the relationship between derivatives and integrals.

The conceptual transition is:

\boxed{ \text{Derivative: local change} \qquad\longleftrightarrow\qquad \text{Integral: accumulated change} }

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Quick Recall & Flashcards

Test your active recall before proceeding to practice problems. Format:

Question :: Answer.

Definition of derivative $f'(x)$ from first principles :: $\lim_{h\to 0} \frac{f(x+h) - f(x)}{h}$
Relationship between differentiability and continuity :: Differentiable $\implies$ Continuous (converse is FALSE)
Power rule for differentiation $\frac{d}{dx}[x^n]$ :: $n x^{n-1}$
Product rule for differentiation $\frac{d}{dx}[f(x)g(x)]$ :: $f'(x)g(x) + f(x)g'(x)$
Quotient rule for differentiation $\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right]$ :: $\frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$
Chain rule for differentiation $\frac{d}{dx}[f(g(x))]$ :: $f'(g(x)) \cdot g'(x)$
Critical point of $f(x)$ definition :: Point in domain where $f'(x) = 0$ or $f'(x)$ is undefined
L'Hôpital's rule condition :: Applicable only to indeterminate forms $\frac{0}{0}$ or $\frac{\pm\infty}{\pm\infty}$

Practice & Self-Check

Attempt these diagnostic questions to verify your operational understanding.

Question 1

Find the derivative of

f(x) = (3x^2 + 2)^4

x = 1

MCQ

3 Unit Assessment

A. 3000

B. 2400

C. 3000

D. 1500

Question 2

If a function

f(x)

is differentiable at

x = c

, does that guarantee that

f(x)

is continuous at

x = c

MCQ

3 Unit Assessment

A. Yes, differentiability strictly implies continuity

B. No, continuity and differentiability are completely unrelated

C. Only if

f'(c) \neq 0

D. Only if

f(x)

is a polynomial