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Derivatives — Definition and Rules

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Derivatives — Definition and Rules

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Derivatives — Definition and Rules

🎯 Learning Objectives

By the end of this topic, you will be able to:

Define the derivative as the limit of the difference quotient
Apply power, product, quotient, and chain rules
Find derivatives of exponential, logarithmic, and trigonometric functions
Use L'Hôpital's rule for limit evaluation
Compute linear approximations using tangent lines

📋 Prerequisites

Limits of Functions — limit definition, L'Hôpital's rule
Exponential & Log Functions — derivatives depend on these

📖 Core Content

13.1 Intuition: Instantaneous Rate of Change

The derivative is the instantaneous rate of change — it tells you how fast a quantity is changing at exactly one moment. If distance

s(t)

describes a car's position,

s'(t)

is the speedometer reading at time

t

. Geometrically, the derivative at

x = a

is the slope of the tangent line to the curve

y = f(x)

at that point.

💡 Why this matters: Derivatives are the mathematical language of change. In ML, gradient descent uses derivatives to minimize loss functions. In physics, derivatives give velocity and acceleration.

13.2 Definition

f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}

Alternative notation:

\frac{dy}{dx}

\frac{d}{dx}f(x)

\dot{y}

(Newton),

f'(x)

y'

. Derivative at a point:

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

(Diagram)

13.3 Basic Derivatives

$f(x)$	$f'(x)$	Notes
$c$ (constant)	$0$
$x^n$	$n x^{n-1}$	Power rule
$e^x$	$e^x$
$a^x$	$a^x \ln a$	$a > 0$
$\ln x$	$\frac{1}{x}$	$x > 0$
$\log_a x$	$\frac{1}{x \ln a}$
$\sin x$	$\cos x$
$\cos x$	$-\sin x$
$\tan x$	$\sec^2 x$
$\sec x$	$\sec x \tan x$

13.4 Differentiation Rules

Rule	Formula	Example
Constant multiple	$(cf)' = cf'$	$(3x^2)' = 6x$
Sum	$(f+g)' = f' + g'$	$(x^2 + x)' = 2x + 1$
Difference	$(f-g)' = f' - g'$
Product	$(fg)' = f'g + fg'$	$(x^2 e^x)' = 2xe^x + x^2 e^x$
Quotient	$\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}$	$\left(\frac{x}{e^x}\right)' = \frac{1\cdot e^x - x\cdot e^x}{e^{2x}}$
Chain	$(f \circ g)' = f'(g(x)) \cdot g'(x)$	$(e^{x^2})' = e^{x^2} \cdot 2x$

13.4.1 Chain Rule — The "Outside-Inside" Rule

\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)

Intuition: Differentiate the outside function (keeping the inside unchanged), then multiply by the derivative of the inside. Examples:

$\frac{d}{dx} \sin(3x) = \cos(3x) \cdot 3 = 3\cos(3x)$
$\frac{d}{dx} e^{x^2} = e^{x^2} \cdot 2x$
$\frac{d}{dx} \ln(x^2 + 1) = \frac{1}{x^2+1} \cdot 2x = \frac{2x}{x^2+1}$

13.5 Worked Examples

Example 1.1 (Power rule): Find

\frac{d}{dx}(3x^5 - 2x^3 + x - 7)

\frac{d}{dx}(3x^5 - 2x^3 + x - 7) = 15x^4 - 6x^2 + 1

Example 1.2 (Product rule): Find

f'(x)

for

f(x) = x^2 \sin x

f'(x) = (2x)(\sin x) + (x^2)(\cos x) = 2x \sin x + x^2 \cos x

Example 1.3 (Quotient rule): Find

g'(x)

for

g(x) = \frac{e^x}{x^2 + 1}

g'(x) = \frac{e^x(x^2+1) - e^x(2x)}{(x^2+1)^2} = \frac{e^x(x^2 - 2x + 1)}{(x^2+1)^2} = \frac{e^x(x-1)^2}{(x^2+1)^2}

Example 1.4 (Chain rule): Find

h'(x)

for

h(x) = \sin(e^{x^2})

. Outside:

\sin

, derivative

\cos(e^{x^2})

. Inside:

e^{x^2}

, derivative

e^{x^2} \cdot 2x

h'(x) = \cos(e^{x^2}) \cdot e^{x^2} \cdot 2x = 2x e^{x^2} \cos(e^{x^2})

13.6 Linear Approximation

The tangent line at

x = a

approximates

f(x)

near

a

L(x) = f(a) + f'(a)(x - a)

Example: Approximate

\sqrt{4.1}

f(x) = \sqrt{x}

a = 4

f(4) = 2

f'(4) = \frac{1}{2\sqrt{4}} = \frac{1}{4}

L(4.1) = 2 + \frac{1}{4}(0.1) = 2 + 0.025 = 2.025

\boxed{\sqrt{4.1} \approx 2.025 \ (\text{actual } \approx 2.0249)}

📐 Key Formulas — Summary Table

Rule	Formula
Definition	$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$
Power	$\frac{d}{dx}x^n = n x^{n-1}$
Product	$(fg)' = f'g + fg'$
Quotient	$(f/g)' = \frac{f'g - fg'}{g^2}$
Chain	$(f \circ g)' = f'(g(x)) \cdot g'(x)$
Exponential	$\frac{d}{dx} e^x = e^x$ , $\frac{d}{dx} a^x = a^x \ln a$
Log	$\frac{d}{dx} \ln x = \frac{1}{x}$
Sin	$\frac{d}{dx} \sin x = \cos x$
Cos	$\frac{d}{dx} \cos x = -\sin x$
Linear approx	$L(x) = f(a) + f'(a)(x-a)$

⚠️ Common Pitfalls

Pitfall 1: Forgetting the Chain Rule

Mistake:

\frac{d}{dx} \sin(5x) = \cos(5x)

(missing the

5

from the chain rule). Correct:

\frac{d}{dx} \sin(5x) = \cos(5x) \cdot 5 = 5\cos(5x)

Pitfall 2: Misapplying the Product Rule

Mistake:

(fg)' = f'g'

(wrong! This is not the product rule). Correct:

(fg)' = f'g + fg'

Pitfall 3: Confusing Quotient Rule Order

Mistake:

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

(swapped numerator order). Correct:

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

. Memory aid: "Low d-high minus high d-low, square the low."

📝 Practice Questions

Q1: Find $f'(x)$ for $f(x) = 4x^3 - 2x^2 + 7x - 5$ .

$f'(x) = 12x^2 - 4x + 7$

$\boxed{12x^2 - 4x + 7}$ Q2: Find $g'(x)$ for $g(x) = x^2 e^x$ .

Product rule: $g' = 2xe^x + x^2 e^x = xe^x(2 + x)$

$\boxed{xe^x(x+2)}$ Q3: Find $\frac{d}{dx} \frac{x^2}{x+1}$ .

Quotient rule: $\frac{2x(x+1) - x^2(1)}{(x+1)^2} = \frac{2x^2+2x-x^2}{(x+1)^2} = \frac{x^2+2x}{(x+1)^2}$

$\boxed{\frac{x(x+2)}{(x+1)^2}}$ Q4: Find $h'(x)$ for $h(x) = (3x^2 + 1)^5$ .

Chain rule: $5(3x^2+1)^4 \cdot 6x = 30x(3x^2+1)^4$

$\boxed{30x(3x^2+1)^4}$ Q5: Find $\frac{d}{dx} \ln(x^2+3)$ .

$\frac{1}{x^2+3} \cdot 2x = \frac{2x}{x^2+3}$

$\boxed{\frac{2x}{x^2+3}}$ Q6: Find $\frac{d}{dx} e^{3x}$ .

$e^{3x} \cdot 3 = 3e^{3x}$

$\boxed{3e^{3x}}$ Q7: Find $\frac{d}{dx} \sin(2x)$ .

$\cos(2x) \cdot 2 = 2\cos(2x)$

$\boxed{2\cos(2x)}$ Q8: Find $f'(1)$ if $f(x) = xe^{-x}$ .

$f'(x) = e^{-x} + x(-e^{-x}) = e^{-x}(1-x)$ $f'(1) = e^{-1}(0) = 0$

$\boxed{0}$ Q9: Approximate $\sqrt[3]{8.1}$ using linear approximation.

$f(x) = \sqrt[3]{x}$ , $a=8$ , $f(8)=2$ , $f'(x)=\frac{1}{3}x^{-2/3}$ , $f'(8)=\frac{1}{3}\cdot\frac{1}{4}=\frac{1}{12}$ $L(8.1) = 2 + \frac{1}{12}(0.1) = 2 + 0.00833 = 2.00833$

$\boxed{2.00833}$ Q10: Find the equation of the tangent line to $f(x) = x^2$ at $x=3$ .

$f(3)=9$ , $f'(x)=2x$ , $f'(3)=6$ $y - 9 = 6(x-3) \implies y = 6x - 9$

$\boxed{y = 6x - 9}$ Q11: Find $\frac{d}{dx} \cos(x^3)$ .

$-\sin(x^3) \cdot 3x^2 = -3x^2 \sin(x^3)$

$\boxed{-3x^2 \sin(x^3)}$ Q12: Find $\frac{d}{dx} \frac{\sin x}{x}$ .

Quotient: $\frac{\cos x \cdot x - \sin x \cdot 1}{x^2} = \frac{x\cos x - \sin x}{x^2}$

$\boxed{\frac{x\cos x - \sin x}{x^2}}$

🔗 Cross-References

Next topic: Applications of Derivatives
Previous: Limits of Functions — definitions
Across courses: BSMA1003 Maths 2 (multivariable derivatives, partial derivatives) Join Discord Previous7.3 ContinuityNext8.2 Applications of Derivatives