Differentiate standard functions

Trigonometric derivatives use radians

Identify the inner function before choosing a derivative. Use radian input, keep the inner derivative as a factor, then check where the original function is defined.

For a differentiable inner function u, all six source rows are:

Function: $\sin u$

• Derivative with respect to x: $\cos u\,u^{\prime}$

Function: $\cos u$

• Derivative with respect to x: $-\sin u\,u^{\prime}$

Function: $\tan u$

• Derivative with respect to x: $\sec^2u\,u^{\prime}$

Function: $\sec u$

• Derivative with respect to x: $\sec u\tan u\,u^{\prime}$

Function: $\csc u$

• Derivative with respect to x: $-\csc u\cot u\,u^{\prime}$

Function: $\cot u$

• Derivative with respect to x: $-\csc^2u\,u^{\prime}$

Setting u=x gives the source’s basic column.

The source spells csc as cosec and shades the basic secant/cosecant rows as listed in MF27.

Preserve the relevant nonzero sine/cosine denominators.

Degrees would introduce an additional angle-conversion factor; these displayed formulas assume radians.

Exponentials and logarithms

The four source rows, including their general inner-function forms, are:

Function: $e^u$

• Derivative with respect to x: $e^u u^{\prime}$

Function: $a^u$

• Derivative with respect to x: $a^u\ln a\,u^{\prime}$

Function: $\ln u$

• Derivative with respect to x: $u^{\prime}/u$

Function: $\log_a u=\ln u/\ln a$

• Derivative with respect to x: $u^{\prime}/(u\ln a)$

The exponential base is positive.

A real logarithm additionally needs u>0 and base a>0, a≠1.

Setting u=x gives every printed basic formula.

Simplifying an answer does not change its original domain: for example, the identity used in Example 7(b) is valid only where its tangent/secant expression exists.