Power, exponential, trig rules

A handful of short rules replace limit computations for the functions you meet most often.

Power rule

For any real number n, the derivative of x^n is n times x^(n-1). The exponent comes down as a multiplier and drops by one. Differentiating a polynomial reduces to applying this rule term by term, since constants multiply through and sums split apart.

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

Exponential functions

The exponential with base e is its own derivative, which is what makes e special: changing the input by a factor multiplies the output by a constant. Other bases carry a factor of the natural logarithm of the base. The general exponential a^x differentiates to a^x times ln a.

\frac{d}{dx} e^x = e^x, \quad \frac{d}{dx} a^x = a^x \ln a

Trig derivatives

The derivative of sin x is cos x, and the derivative of cos x is negative sin x; the two cycle with a period of four differentiations. The derivative of tan x is secant squared of x. Combine these with the power rule term by term for mixed functions.

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

Flashcard

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

Power rule for any real exponent

Bring the exponent down as a coefficient and reduce it by one. It works for negative and fractional exponents too, so roots and reciprocals need rewriting before differentiating.

Check your understanding

What is the derivative of f(x) = x^5 - 3 \cos x?

Take the unit quiz

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