Unit content
Calculating derivatives
The derivative definition can be used directly, but most derivatives are computed efficiently from a small set of rules.
Linearity and powers
Derivatives distribute over sums and constant multiples:
$$(af+bg)'=af'+bg'.$$
For a power,
$$\frac{d}{dx}x^n=nx^{n-1}$$
where the rule is valid on the appropriate domain.
Products and quotients
If two differentiable functions are multiplied,
$$(fg)'=f'g+fg'.$$
For a quotient with $g\ne0$,
$$\left(\frac{f}{g}\right)'=\frac{f'g-fg'}{g^2}.$$
The derivative of a product is therefore not generally the product of the derivatives.
The chain rule
When one function is composed with another,
$$(f\circ g)'(x)=f'(g(x))g'(x).$$
For example,
$$\frac{d}{dx}(3x+1)^4 =4(3x+1)^3\cdot3 =12(3x+1)^3.$$
Standard derivatives
Together with these structural rules, familiar function families have standard derivatives. Important examples include
$$\frac{d}{dx}e^x=e^x,$$
$$\frac{d}{dx}\ln x=\frac1x\qquad(x>0),$$
$$\frac{d}{dx}\sin x=\cos x,$$
$$\frac{d}{dx}\cos x=-\sin x.$$
Most elementary derivative calculations reduce a complicated expression to repeated applications of these basic patterns.