## Chapter 2

Differentiation

The concept of the derivative is one of the two fundamental ideas of calculus, and differentiation is the process of finding the derivative of a function. The derivative of a function is the slope of its tangent line to its graph, and as we will see in this chapter, it is a mathematical tool to measure the rate of change of one quantity relative to another.

The graph of a function \(f\) is shown in Figure 2.0.1. Give a rough sketch of the graph of \(f’.\)

Solution 2. The tangent line to the curve is horizontal at \(x=0\) and \(x=2.\) \[ \begin {cases} x<0 & \text {tangent line makes an acute angle with positive \ensuremath {x} axis}\Rightarrow f'(x)>0\\ 0<x<2 & \text {tangent line makes an obtuse angle with positive \ensuremath {x} axis}\Rightarrow f'(x)<0\\ 2<x & \text {tangent line makes an acute angle with positive \ensuremath {x} axis}\Rightarrow f'(x)>0 \end {cases} \] Specifically, the slope of the tangent line at \(x=-1\) is \[ m_{\text {tan}}=\frac {\text {rise}}{\text {run}}=\frac {3}{1}\Rightarrow f'(-1)=3. \] The slope of the tangent line at \(x=1\) is \[ m_{\text {tan}}=\frac {\text {rise}}{\text {run}}=\frac {-1}{1}\Rightarrow f'(1)=-1. \] At \(x=3\) \[ m_{\text {tan}}=\frac {\text {rise}}{\text {run}}=\frac {-3}{-1}\Rightarrow f'(3)=3. \]