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Section 2.1 Introduction

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


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.