Question

Use the differentiation rules developed in this section to find the derivatives of the functions in Exercises 35-64. Note that it may be necessary to do some preliminary algebra before differentiating.

$f\left(x\right)=\frac{x^4-7x^3}{2x}$

Short answer

The derivative of the function is$\begin{array}{rcl}f\text{'}\left(x\right)& =& \frac{3}{2}{x}^2-7x\end{array}$.

Step-by-step solution

Step 1. Given Information

The given function is$f\left(x\right)=\frac{x^4-7x^3}{2x}$.

Step 2. Simplify the function

Simplify the given function.

$\begin{array}{rcl}f\left(x\right)& =& \frac{x^4}{2x}-\frac{7x^3}{2x}\\ & =& \frac{x^3}{2}-\frac{7}{2}{x}^2\end{array}$

Step 3. Find the derivative

• Apply the difference rule of derivative, $(f-g)\text{'}\left(x\right)=f\text{'}\left(x\right)-g\left(x\right)$.

$\begin{array}{rcl}f\text{'}\left(x\right)& =& \frac{d}{dx}\left(\frac{x^3}{2}\right)-\frac{d}{dx}(\frac{7}{2}{x}^2)\end{array}$

• Apply the constant multiple rule of derivative, localid="1648555871956" $\left(kf\right)\text{'}\left(x\right)=kf\text{'}\left(x\right)$.

$f\text{'}\left(x\right)=\frac{1}{2}\begin{array}{rcl}& & \frac{d}{dx}\left(x^3\right)-\frac{7}{2}\frac{d}{dx}(x^2)\end{array}$

• Apply the power rule of derivative, localid="1648547791959" $\left(x^n\right)\text{'}=n{x}^{n-1}$.

localid="1648547849761" $\begin{array}{rcl}f\text{'}\left(x\right)& =& \frac{1}{2}\left(3x^2\right)-\frac{7}{2}(2x)\\ & =& \frac{3}{2}{x}^2-7x\end{array}$