Derivatives of Sine, Cosine, e^x, and ln x

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Key Ideas Easy Medium Hard

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

e^x stays e^x. The derivative of ln x is 1 over x.

What to Remember

Four derivatives to know

\(\dfrac{d}{dx}\sin x = \cos x\)

\(\dfrac{d}{dx}\cos x = -\sin x\)

\(\dfrac{d}{dx} e^x = e^x\)

\(\dfrac{d}{dx}\ln x = \dfrac{1}{x}\)

A constant multiple rides along

The derivative of \(5\sin x\) is \(5\cos x\). The derivative of \(-2\cos x\) is \(2\sin x\).

Add the derivatives

For a sum, differentiate each term. A polynomial term still uses the power rule.

Evaluate after you differentiate

Find \(f'(x)\) on its own line. Then substitute the angle or the number.

Level 1

Easy

One function, or a sum of two.

Example

Find \(f'(x)\) for \(f(x) = 3\sin x\).

\(\dfrac{d}{dx}\sin x = \cos x\)

\(f'(x) = 3\cos x\)

Answer: \(3\cos x\)

Find \(f'(x)\) for \(f(x) = 5\cos x\).

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\(\dfrac{d}{dx}\cos x = -\sin x\)

\(f'(x) = -5\sin x\)

Answer: \(-5\sin x\)

Find \(f'(x)\) for \(f(x) = 4e^x\).

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\(\dfrac{d}{dx} e^x = e^x\)

\(f'(x) = 4e^x\)

Answer: \(4e^x\)

Find \(f'(x)\) for \(f(x) = 7\ln x\).

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\(\dfrac{d}{dx}\ln x = \dfrac{1}{x}\)

\(f'(x) = \dfrac{7}{x}\)

Answer: \(\dfrac{7}{x}\)

Find \(f'(x)\) for \(f(x) = \sin x + x^2\).

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\(\dfrac{d}{dx}\sin x = \cos x\)

\(\dfrac{d}{dx} x^2 = 2x\)

\(\cos x + 2x\)

Answer: \(\cos x + 2x\)

Find \(f'(x)\) for \(f(x) = e^x - \cos x\).

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\(\dfrac{d}{dx} e^x = e^x\)

\(\dfrac{d}{dx}(-\cos x) = \sin x\)

\(e^x + \sin x\)

Answer: \(e^x + \sin x\)

Level 2

Medium

Differentiate, then substitute a point.

Example

For \(f(x) = \sin x\), find \(f'\!\bigl(\tfrac{\pi}{3}\bigr)\).

\(f'(x) = \cos x\)

\(\cos\!\bigl(\tfrac{\pi}{3}\bigr) = \dfrac{1}{2}\)

Answer: \(\dfrac{1}{2}\)

For \(f(x) = \cos x\), find \(f'\!\bigl(\tfrac{\pi}{6}\bigr)\).

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\(f'(x) = -\sin x\)

\(-\sin\!\bigl(\tfrac{\pi}{6}\bigr) = -\dfrac{1}{2}\)

Answer: \(-\dfrac{1}{2}\)

For \(f(x) = \ln x\), find \(f'(4)\).

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\(f'(x) = \dfrac{1}{x}\)

\(f'(4) = \dfrac{1}{4}\)

Answer: \(\dfrac{1}{4}\)

For \(f(x) = e^x + 2x\), find \(f'(0)\).

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\(f'(x) = e^x + 2\)

Substitute \(x = 0\).

\(e^0 + 2 = 1 + 2 = 3\)

Answer: \(3\)

For \(f(x) = 3\cos x\), find \(f'(\pi)\).

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\(f'(x) = -3\sin x\)

\(-3\sin \pi = 0\)

Answer: \(0\)

For \(f(x) = e^x + \ln x\), find \(f'(1)\).

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\(f'(x) = e^x + \dfrac{1}{x}\)

Substitute \(x = 1\).

\(e^1 + 1 = e + 1\)

Answer: \(e + 1\)

Level 3

Hard

A sum of three, or a choice that hides a sign error.

Example

For \(f(x) = x^2 + \sin x + e^x\), find \(f'(0)\).

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

Substitute \(x = 0\).

\(0 + \cos 0 + e^0 = 1 + 1 = 2\)

Answer: \(2\)

For \(f(x) = 4\ln x - \cos x\), find \(f'\!\bigl(\tfrac{\pi}{2}\bigr)\).

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\(f'(x) = \dfrac{4}{x} - (-\sin x) = \dfrac{4}{x} + \sin x\)

Substitute \(x = \tfrac{\pi}{2}\).

\(\dfrac{4}{\pi/2} + \sin\!\bigl(\tfrac{\pi}{2}\bigr) = \dfrac{8}{\pi} + 1\)

Answer: \(1 + \dfrac{8}{\pi}\)

Which expression is the derivative of \(6e^x - 2\sin x\)?

A. \(6e^x - 2\cos x\)

B. \(6xe^{x - 1} - 2\cos x\)

C. \(6e^x + 2\cos x\)

D. \(e^x - \cos x\)

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\(\dfrac{d}{dx}(6e^x) = 6e^x\)

\(\dfrac{d}{dx}(-2\sin x) = -2\cos x\)

\(6e^x - 2\cos x\)

Answer: A

For \(f(x) = \ln x + \cos x\), find \(f'(\pi)\).

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\(f'(x) = \dfrac{1}{x} - \sin x\)

Substitute \(x = \pi\).

\(\dfrac{1}{\pi} - \sin \pi = \dfrac{1}{\pi}\)

Answer: \(\dfrac{1}{\pi}\)

The derivative of \(\ln x\) at \(x = e\) is

A. \(1\)

B. \(0\)

C. \(\dfrac{1}{e}\)

D. \(e\)

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\(\dfrac{d}{dx}\ln x = \dfrac{1}{x}\)

Substitute \(x = e\).

\(\dfrac{1}{e}\)

Answer: C. \(\dfrac{1}{e}\)

For \(f(x) = 2e^x - \ln x + \sin x\), find \(f'(1)\).

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\(f'(x) = 2e^x - \dfrac{1}{x} + \cos x\)

Substitute \(x = 1\).

\(2e - 1 + \cos 1\)

Answer: \(2e - 1 + \cos 1\)

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