Differentiation Review

Many functions can't be cracked open with a single rule. Instead, we can use a rule to break the problem down into (possibly several) simpler problems. Then we can use another rule on each of those. And repeat as long as needed to get the answer.

Example: Find

$F'(x)$ where

$F(x)=\frac{\sin\left(x^2\right)}{1 + e^{2x}}.$
Solution: This is a ratio, so

$F'(x) = \frac{\left(1+e^{2x}\right)\frac{d}{dx}\Big((\sin\left(x^2\right)\Big) - \sin\left(x^2\right)\frac{d}{dx}\Bigl(1+e^{2x}\Bigr)}{\left(1+e^{2x}\right)^2},$ by the quotient rule. But that still leaves us with the question of how to compute the derivatives of

$\sin\left(x^2\right)$ and

$\left(1+e^{2x}\right)$ . Each of those can be computed using the chain rule.

It's very dangerous to combine steps! Until you've really got the hang of it, write out your calculations with at most one rule per line. For instance, you might write:
$\begin{eqnarray*}F'(x) = & \frac{(1+e^{2x})\frac{d}{dx}\Big(\sin\left(x^2\right)\Big) - \sin\left(x^2\right)\frac{d}{dx}\Bigl(1+e^{2x}\Bigr)}{\left(1+e^{2x}\right)^2} & \qquad \hbox{(by the quotient rule)} \cr= & \frac{\left(1+e^{2x}\right)\left(\cos\left(x^2\right)\right)(2x) - \sin\left(x^2\right)\frac{d}{dx}\Bigl(1+e^{2x}\Bigr)}{\left(1+e^{2x}\right)^2} & \qquad \hbox{(by the chain rule applied to }\sin\left(x^2\right)\hbox{)} \cr= & \frac{\left(1+e^{2x}\right)\left(\cos\left(x^2\right)\right)(2x) - \sin\left(x^2\right)e^{2x}\frac{d}{dx}\Bigl(2x\Bigr)}{\left(1+e^{2x}\right)^2} & \qquad \hbox{(by the chain rule applied to }e^{2x}\hbox{)} \cr= & \frac{2x\left(1+e^{2x}\right)\cos(x^2) - 2\sin\left(x^2\right)e^{2x}}{\left(1+e^{2x}\right)^2} & \qquad \hbox{(by algebra)} \cr\end{eqnarray*}$

Eventually you'll get the hang of it to the point that you can do several operations in one line, but please be patient.

There isn't always an obvious order in which to apply the rules. When in doubt, think about the structure of the function. Would you describe it as a product of two simpler functions? If so, apply the product rule first! Would you describe it as a quotient? If so, apply the quotient rule first. Is it instead a power, or a log, or an exponential, or a trig function of some complicated expression (which may itself involve products, quotients, or further nesting)? If so, apply the chain rule first.

The Six Pillars of Calculus

Trigonometry Review

Exponential Functions

Logarithms and Inverse functions

Intro to Limits

Limit Laws and Computations

Continuity and the Intermediate Value Theorem

Limits at Infinity

Rates of Change

The Derivative Function

Basic Differentiation Rules

Product and Quotient Rules

Derivatives of Trig Functions

The Chain Rule

Implicit Differentiation

Derivatives of Logs

Derivatives in Science

Related Rates

Linear Approximation and Differentials

Differentiation Review

Absolute and Local Extrema

The Mean Value and other Theorems

ff vs. f′f'

Indeterminate Forms and L'Hospital's Rule

Optimization

Newton's Method

Anti-derivatives

The Area under a curve

Definite Integrals

The Fundamental Theorem of Calculus

Combining Rules

$f$ vs. $f'$