Math · Calculus I · Concept

Derivatives of eˣ, ln x and trig functions

The derivative of eˣ is eˣ, the derivative of ln x is 1/x for x > 0, the derivative of sin x is cos x and the derivative of cos x is −sin x, with x in radians. Each follows from the limit definition, and each combines with the sum, product and chain rules.

eˣ is its own derivative

For a base b > 0, the difference quotient of bˣ factors as bˣ·(bʰ − 1)/h, so the derivative of bˣ is bˣ times a constant that depends only on b. The number e ≈ 2.71828 is the base that makes that constant exactly 1, so eˣ grows at a rate equal to its own value. For other bases the constant is ln b.

\frac{d}{d x} e^{x} = e^{x}, \frac{d}{d x} b^{x} = b^{x} ln b

\begin{array}{c} \frac{d}{d x} e^{x} = e^{x} \\ \frac{d}{d x} b^{x} = b^{x} ln b \end{array}

ln x has derivative 1/x

ln x undoes eˣ: e^(ln x) = x for x > 0. Differentiating both sides with the chain rule gives e^(ln x)·(ln x)′ = 1, and since e^(ln x) = x, (ln x)′ = 1/x. The formula holds where ln x is defined, x > 0.

\frac{d}{d x} ln x = \frac{1}{x} (x > 0)

What about other bases and negative x?

log_b x = ln x / ln b, so its derivative is 1/(x ln b). For x ≠ 0, ln|x| has derivative 1/x on both sides of zero.

Sine and cosine

Putting the sine addition formula into the difference quotient leaves two standard limits: sin h/h → 1 and (cos h − 1)/h → 0 as h → 0. They give (sin x)′ = cos x, and the same argument gives (cos x)′ = −sin x. Both limits, and so both formulas, need x in radians.

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

\begin{array}{c} \frac{d}{d x} sin x = cos x \\ \frac{d}{d x} cos x = - sin x \end{array}

The other trigonometric functions

Write tan x = sin x / cos x and apply the quotient rule to get sec²x. The same approach gives the derivatives of sec x, csc x and cot x.

Derivatives of the other trigonometric functions
f(x)	f′(x)
tan x	sec²x
sec x	sec x tan x
csc x	−csc x cot x
cot x	−csc²x

Combine them with the other rules

These formulas are building blocks. Constant multiples and sums follow the usual rules, a product such as eˣ sin x needs the product rule, and a function of a function such as e^(3x) or sin(x²) needs the chain rule.

\frac{d}{d x} e^{3 x} = 3 e^{3 x}, \frac{d}{d x} sin (x^{2}) = 2 x cos (x^{2})

\begin{array}{c} \frac{d}{d x} e^{3 x} = 3 e^{3 x} \\ \frac{d}{d x} sin (x^{2}) = 2 x cos (x^{2}) \end{array}

Common mistakes

Using the power rule on eˣ: eˣ is not a power of x, and its derivative is eˣ, not x·eˣ⁻¹.
Working in degrees: with x in degrees, (sin x)′ = (π/180)cos x, so the standard formulas assume radians.
Getting the sign of the cosine derivative wrong: (cos x)′ = −sin x.
Forgetting the chain rule inside ln or e: (ln 5x)′ = 5/(5x) = 1/x, and (e^(2x))′ = 2e^(2x).

Key terms

Euler’s number: The irrational constant e ≈ 2.71828, the base of natural exponentials and logarithms. The function eˣ is its own derivative: its slope always equals its value.
Exponential function: A function with the variable in the exponent, such as bˣ with a fixed base b > 0, b ≠ 1. The natural exponential eˣ is its own derivative.
Natural logarithm: The logarithm with base e, written ln x. It undoes eˣ: ln(eˣ) = x for every x, and e^(ln x) = x for x > 0.
Radian: An angle measure: arc length divided by radius. A full turn is 2π radians (360°), and calculus formulas such as (sin x)′ = cos x assume radians.
Derivative: The instantaneous rate of change of a function: the limit of the average rate of change as the step shrinks to zero, when that limit exists. On a graph it is the slope of the tangent line.
Chain rule: The rule for differentiating a function inside another function: differentiate the outside, keeping the inside as it is, then multiply by the derivative of the inside.

Work through an example

Differentiate f(x) = 3eˣ − 2 ln x + 5 sin x, then find the slope of its graph at x = 1 (x in radians).

Differentiate eˣ, ln x and sin x terms →

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