## Section2.1Introduction

Use the following definitions, techniques, properties, and algorithms to solve the problems contained in this Chapter.

Derivative

The derivative of a function $f$ at a number $a$ is $\displaystyle f^\prime(a)=\lim _{h\to 0}\frac{f(a+h)-f(a)}{h}$ if this limit exists.

Tangent Line

An equation of the tangent line to $y=f(x)$ at $(a,f(a))$ is given by $y-f(a)=f'(a)(x-a)\text{.}$

Product and Quotient Rules

If $f$ and $g$ are both differentiable, then $(fg)' = f\cdot g' +g \cdot f'$ and $\displaystyle \left(\frac{f}{g}\right)' = \frac{g\cdot f' -f \cdot g'}{g^2}\text{,}$ with $g(x)\not= 0\text{.}$

Chain Rule

If $f$ and $g$ are both differentiable and $F=f\circ g$ is the composite function defined by $F(x)=f(g(x))\text{,}$ then $F$ is differentiable and $F'$ is given by $F'(x)=f'(g(x))\cdot g'(x)\text{.}$

Implicit Differentiation

Let a function $y=y(x)$ be implicitly defined by $F(x,y)=G(x,y)\text{.}$ To find the derivative $y'$ do the following:

1. Use the chain rule to differentiate both sides of the given equation, thinking of $x$ as the independent variable.

2. Solve the resulting equation for $\ds \frac{dy}{dx}\text{.}$

The Method of Related Rates

If two variables are related by an equation and both are functions of a third variable (such as time), we can find a relation between their rates of change. We say the rates are related, and we can compute one if we know the other. We proceed as follows:

1. Identify the independent variable on which the other quantities depend and assign it a symbol, such as $t\text{.}$ Also, assign symbols to the variable quantities that depend on $t\text{.}$

2. Find an equation that relates the dependent variables.

3. Differentiate both sides of the equation with respect to $t$ (using the chain rule if necessary).

4. Substitute the given information into the related rates equation and solve for the unknown rate.