Calculus

Introduction to derivatives

The derivative as a rate of change and the slope of a tangent, and the rules that make it computable.

About 12 minutes

The derivative answers the question “how fast is this quantity changing right now?” Geometrically it is the slope of the tangent line to a graph; physically it is velocity when the function is position, or marginal cost when the function is cost.

From average to instantaneous rate

The average rate of change of $f$ between $x$ and x + h is the slope of the secant line through the two points. Letting $h$ shrink to zero turns the secant into the tangent, and the limit of the slopes — when it exists — is the derivative.

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

The basic rules

Computing limits every time would be slow, so derivatives are found with rules. The power rule handles powers of $x$ ; constants multiply through and sums are differentiated term by term.

Other building blocks: \frac{d}{dx} e^x = e^x, \frac{d}{dx} \ln x = \frac{1}{x}, \frac{d}{dx} \sin x = \cos x and \frac{d}{dx} \cos x = -\sin x.

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

Products, quotients and compositions

The derivative of a product is not the product of the derivatives. The product and quotient rules give the correct combinations, and the chain rule differentiates a function of a function: differentiate the outer function, keep the inner one in place, then multiply by the derivative of the inner function.

(fg)' = f'g + fg',\qquad \frac{d}{dx} f(g(x)) = f'(g(x))\,g'(x)

What the derivative tells you

Where f'(x) > 0 the function is increasing; where f'(x) < 0 it is decreasing. Points where f'(x) = 0 are candidates for local maxima and minima, and the second derivative describes how the graph bends: positive means concave up.

Worked examples

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

Power rule on each term: 5 \cdot 4x^3 = 20x^3, and -2x becomes $-2$ .
The constant 7 has derivative 0.
f'(x) = 20x^3 - 2.

Answer: 20x^3 - 2

Differentiate y = \sin(x^2).

Outer function \sin u, inner function u = x^2.
Derivative of the outer: \cos u. Derivative of the inner: $2x$ .
Chain rule: y' = \cos(x^2) \cdot 2x.

Answer: 2x\cos(x^2)

Common mistakes

Differentiating a product factor by factor: (x^2 \sin x)' \ne 2x \cos x.
Forgetting to multiply by the derivative of the inner function in the chain rule.
Applying the power rule to e^x or 2^x, where the variable is in the exponent.
Treating the derivative of a constant as the constant itself rather than 0.

Practice problems

1.\frac{d}{dx} x^3
2.\frac{d}{dx} (x^2 \sin x)
3.\frac{d}{dx} e^{3x}
4.Slope of y = x^2 at x = 3

Practise this topic with checked answers

Tools for this lesson

Derivative CalculatorDifferentiate functions symbolically — first, second and higher derivatives — with the rules used and a numerical verification.Limit CalculatorFinite, one-sided and infinite limits, including 0/0 and ∞/∞ forms, estimated numerically and confirmed symbolically when possible.Graphing CalculatorPlot and compare functions, implicit equations and inequalities with zoom, pan, coordinate tracking and PNG export.Numerical Differentiation CalculatorApproximate derivatives with forward, backward, central and five-point differences and compare with the symbolic value.

Next lesson: Introduction to integrals

From average to instantaneous rate

The average rate of change of

f

between

x

and

x + h

x + h is the slope of the secant line through the two points. Letting

h

shrink to zero turns the secant into the tangent, and the limit of the slopes — when it exists — is the derivative.

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

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

The basic rules

Computing limits every time would be slow, so derivatives are found with rules. The power rule handles powers of

x

; constants multiply through and sums are differentiated term by term.

Other building blocks:

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

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

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

\frac{d}{dx} \ln x = \frac{1}{x},

\frac{d}{dx} \sin x = \cos x

\frac{d}{dx} \sin x = \cos x and

\frac{d}{dx} \cos x = -\sin x

\frac{d}{dx} \cos x = -\sin x.

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

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

Products, quotients and compositions

(fg)' = f'g + fg',\qquad \frac{d}{dx} f(g(x)) = f'(g(x))\,g'(x)

(fg)' = f'g + fg',\qquad \frac{d}{dx} f(g(x)) = f'(g(x))\,g'(x)

What the derivative tells you

Where

f'(x) > 0

f'(x) > 0 the function is increasing; where

f'(x) < 0

f'(x) < 0 it is decreasing. Points where

f'(x) = 0

f'(x) = 0 are candidates for local maxima and minima, and the second derivative describes how the graph bends: positive means concave up.

Worked examples

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

Power rule on each term: $5 \cdot 4x^3 = 20x^3$ 5 \cdot 4x^3 = 20x^3, and $-2x$ -2x becomes $-2$ .
The constant 7 has derivative 0.
$f'(x) = 20x^3 - 2$ f'(x) = 20x^3 - 2.

Answer: $20x^3 - 2$ 20x^3 - 2

Differentiate $y = \sin(x^2)$ y = \sin(x^2).

Outer function $\sin u$ \sin u, inner function $u = x^2$ u = x^2.
Derivative of the outer: $\cos u$ \cos u. Derivative of the inner: $2x$ .
Chain rule: $y' = \cos(x^2) \cdot 2x$ y' = \cos(x^2) \cdot 2x.

Answer: $2x\cos(x^2)$ 2x\cos(x^2)

Common mistakes

Differentiating a product factor by factor:

(x^2 \sin x)' \ne 2x \cos x

(x^2 \sin x)' \ne 2x \cos x.

Forgetting to multiply by the derivative of the inner function in the chain rule.

Applying the power rule to

e^x

e^x or

2^x

2^x, where the variable is in the exponent.

Treating the derivative of a constant as the constant itself rather than 0.

Practice problems

\frac{d}{dx} x^3

\frac{d}{dx} x^3

\frac{d}{dx} (x^2 \sin x)

\frac{d}{dx} (x^2 \sin x)

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

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

4.Slope of

y = x^2

y = x^2 at

x = 3

x = 3

Practise this topic with checked answers