Special Derivatives

(e^x)' = e^x, ( x)' = 1/x, ( )' = , ( )' = - , ( )' = ^2.

The idea

Five functions appear so often in calculus that their derivatives should be known by heart: $\frac{d}{dx}e^{x} = e^{x}, \qquad \frac{d}{dx}\ln x = \frac{1}{x},$ $\frac{d}{dx}\sin x = \cos x, \qquad \frac{d}{dx}\cos x = -\sin x, \qquad \frac{d}{dx}\tan x = \sec^{2} x.$ The power rule does not produce these: none of the five is a polynomial, so each derivative must be established separately and remembered.

None of them is arbitrary. The base $e$ is chosen exactly so that $e^{x}$ grows at a rate equal to its own value; every other exponential differentiates to itself times a constant factor. The function $\ln x$ is the inverse of $e^{x}$, so its graph is the same curve reflected across the line $y = x$; the reflection swaps rise with run, turning each slope into its reciprocal, and this is why $\ln x$ differentiates to $1/x$. Differentiating sine and cosine advances each by a quarter-turn: $\sin$ becomes $\cos$, $\cos$ becomes $-\sin$, and four differentiations return to the start. The last entry follows from the others: $\tan x = \sin x / \cos x$, so the quotient rule applies, and $\sin^{2} x + \cos^{2} x = 1$ collapses the result to $1/\cos^{2} x$, written $\sec^{2} x$.

Ways to work on it

• Walkthrough. The five derivatives every calculus student should know.
• Practice. Pick the derivative of one of the standard functions.
• Hardest. Differentiate composites of the standard functions with the chain rule.

Not sure where to start? Take the ten-question placement test.