Chapter 11.3, Problem 31E

To determine

To calculate:

The derivative of the following function:

  $f\left(x\right)=2x^2-7x$

Answer to Problem 31E

The derivative of $f\left(x\right)=2x^2-7x$ is $4x-7$ .

Explanation of Solution

Given information:

  $f\left(x\right)=2x^2-7x$

Formula used:

The derivate of function is given by the following expression:

  $f\text{'}\left(x\right)=\underset{h\to 0}{\lim }\frac{h\left(x+h\right)-h\left(x\right)}{h}$

Calculation:

The given function is:

  $f\left(x\right)=2x^2-7x$

The derivative of $f\left(x\right)=2x^2-7x$ is calculated as:

  $\begin{array}{l}f\text{'}\left(x\right)=\underset{h\to 0}{\lim }\frac{f\left(x+h\right)-f\left(x\right)}{h}\\ f\text{'}\left(x\right)=\underset{h\to 0}{\lim }\frac{2{\left(x+h\right)}^2-7\left(x+h\right)-2x^2+7x}{h}\end{array}$

  $\begin{array}{l}f\text{'}\left(x\right)=\underset{h\to 0}{\lim }\frac{2x^2+4xh+h^2-7x-7h-2x^2+7x}{h}\\ f\text{'}\left(x\right)=\underset{h\to 0}{\lim }\frac{4xh+h^2-7h}{h}\\ f\text{'}\left(x\right)=\underset{h\to 0}{\lim }\frac{h\left(4x+h-7\right)}{h}\\ f\text{'}\left(x\right)=\underset{h\to 0}{\lim }4x+h-7\\ f\text{'}\left(x\right)=\underset{h\to 0}{\lim }4x+0-7\left[\because h=0\right]\\ f\text{'}\left(x\right)=4x-7\\ \end{array}$