General rules

Here are some general rules we know. We will assume that when more than one function is specified that the functions have the same domain. In each case it is part of the conclusion that the new function has a derivative and that it is given by the formula.

Sums:

If f and g are differentiable at a then (f+g)'(a) = f'(a) + g'(a).

$$(\sin + \cos)'(a) = \cos(a) + (-\sin(a))$$

Products:

If f and g are differentiable at a then $(f\cdot g)'(a) = f(a)g'(a) + f'(a)g(a)$.

$$(\sin\cdot\cos)'(a) = \sin(a)(-\sin(a)) + \cos(a)\cos(a)$$

Quotients:

If f and g are differentiable at a and (f/g) is defined at a then $${(f/g)'(a) = \frac{f'(a)g(a)-f(a)g'(a)}{(g(a))^2}}$$.

$$\left(\frac{\sin}{\cos}\right)'(a) = \frac{\cos(a)\cos(a)-\sin(a)(-\sin(a))}{(\cos(a))^2}$$

Compositions:

If f is differentiable at a and g is differentiable at f(a) then $(g\circ f)'(a) = g'(f(a))f'(a)$.

$$(\sin\circ\cos)'(a) = \cos(\cos(a))(-\sin(a))$$

Inverses:

If I is an open interval, $f:I\rightarrow(-\infty,\infty)$has inverse function f^-1, f is continuous on I and $f'(f^{-1}(a)) \neq 0$ then $${(f^{-1})'(a) = \frac{1}{f'(f^{-1}(a))}}$$.

$$\ln'(a) = \frac{1}{e^{\ln(a)}}$$