Derivatives

Derivatives of elementary functions

The derivative gives a tangent slope at every allowed input. Move the contact point and connect each familiar function to its derivative formula.

Linear, quadratic, polynomial, rational, exponential, logarithmic, and trigonometric functions.

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1. Linear functions

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Example graph

f(x)=2x+1,\quad f'(x)=2

A line f(x) = mx + b has derivative m. Its tangent is the line itself, so changing the contact point never changes the slope. A constant function has slope zero.

a: 1
f(a): 3
f′(a): 2

The solid curve is f(x); the dashed line is its tangent at a. The derivative f′(a) is the slope of that line, not the height of the curve. Questions use their stated inputs.

2. Quadratic functions

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Example graph

f(x)=x^2,\quad f'(x)=2x

For f(x) = x², the derivative is 2x. The tangent slopes down at negative inputs, becomes horizontal at zero, and slopes up at positive inputs.

a: 1
f(a): 1
f′(a): 2

The solid curve is f(x); the dashed line is its tangent at a. The derivative f′(a) is the slope of that line, not the height of the curve. Questions use their stated inputs.

3. Polynomial functions

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Example graph

f(x)=x^3-x,\quad f'(x)=3x^2-1

For positive integers n, the power rule gives (xⁿ)′ = nxⁿ⁻¹. Apply it term by term to x³ − x to obtain 3x² − 1. Where this derivative is zero, the tangent is horizontal.

a: 1
f(a): 0
f′(a): 2

The solid curve is f(x); the dashed line is its tangent at a. The derivative f′(a) is the slope of that line, not the height of the curve. Questions use their stated inputs.

4. Rational functions

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Example graph

f(x)=\frac{1}{x},\quad f'(x)=-\frac{1}{x^2}

For f(x) = 1/x, the derivative is −1/x² for x ≠ 0. The function decreases on each side of its excluded input. Explore the positive branch here; moving toward zero makes its tangent steeper.

a: 1
f(a): 1
f′(a): -1

The solid curve is f(x); the dashed line is its tangent at a. The derivative f′(a) is the slope of that line, not the height of the curve. Questions use their stated inputs.

5. Exponential functions

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Example graph

f(x)=e^x,\quad f'(x)=e^x

The natural exponential is its own derivative: (eˣ)′ = eˣ. Its tangent slope equals its height. More generally, (bˣ)′ = bˣ ln(b) for b > 0.

a: 1
f(a): 2.718
f′(a): 2.718

The solid curve is f(x); the dashed line is its tangent at a. The derivative f′(a) is the slope of that line, not the height of the curve. Questions use their stated inputs.

6. Logarithmic functions

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Example graph

f(x)=\ln(x),\quad f'(x)=\frac{1}{x}

For the natural logarithm, (ln x)′ = 1/x on x > 0. It keeps increasing, but its positive tangent slope becomes smaller as x grows. For base b > 0, b ≠ 1, (log_b x)′ = 1/(x ln b).

a: 1
f(a): 0
f′(a): 1

The solid curve is f(x); the dashed line is its tangent at a. The derivative f′(a) is the slope of that line, not the height of the curve. Questions use their stated inputs.

7. Trigonometric functions

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Example graph

f(x)=\sin(x),\quad f'(x)=\cos(x)

With angles in radians, (sin x)′ = cos x and (cos x)′ = −sin x. Move along the sine curve: its tangent is horizontal at a peak or trough. Also, (tan x)′ = 1/cos²x wherever cos x ≠ 0.

a: 1
f(a): 0.841
f′(a): 0.54

The solid curve is f(x); the dashed line is its tangent at a. The derivative f′(a) is the slope of that line, not the height of the curve. Questions use their stated inputs.