Differentiation and Integration

It is expected that you can differentiate and integrate functions such as $\exp(x)$, $\sin(x)$, $\cos(x)$, and polynomials $x^n$, $1/x^n$. You should be familiar with the product rule,

$$\frac{d}{dx}(fg) = \frac{df}{dx}g+f\frac{dg}{dx},$$

the quotient rule,

$$\frac{d}{dx}\left(\frac{f}{g}\right) = \frac{\frac{df}{dx}g-f\frac{dg}{dx}}{g^2},$$

and the chain rule of differentiation,

$$\frac{d}{dx}(f(g(x))) = \frac{df}{dx}(g(x))\frac{dg}{dx}(x),$$

Some derivatives:

$\frac{d}{dx} x^n = nx^{n-1}$   $\frac{d}{dx} \ln(x) = \frac{1}{x}$
$\frac{d}{dx} \sin(cx) = c\cos(cx)$ $\frac{d}{dx} \cos(cx) = -c\sin(cx)$
$\frac{d}{dx} \exp(cx) = c\exp(cx)$ 

$f(x)=$  
$\frac{df}{dx} = $

Some integrals:

$\int\limits_a^b x^n dx = \frac{1}{n}\left(a^{n+1}-b^{n+1}\right)$   $\int\limits_a^b \frac{dx}{x} = \ln\left(\frac{a}{b}\right)$
$\int\limits_a^b \sin(cx) dx= -\frac{\cos(ca)-\cos(cb)}{c}$ $\int\limits_a^b \cos(cx) dx= \frac{\sin(ca)-\sin(cb)}{c}$
$\int\limits_a^b \exp(cx) dx= \frac{\exp(ca)-\exp(cb)}{c}$ 
Questions