Question

Find the first and second derivatives of the function. Check to see that your answers are reasonable by comparing the graphs of \(f\), \(f'\), and \(f''\).

51. \(f\left( x \right) = 2x - 5{x^{3/4}}\)

Short answer

The first derivative is \(f'\left( x \right) = 2 - \frac{{15}}{4}{x^{ - 1/4}}\)and the second derivative is \(f''\left( x \right) = \frac{{15}}{{16}}{x^{ - 5/4}}\).

From the graph, as \(f\)is positive and is decreasing in nature, then \(f'\)is negative and is always increasing. As \(f'\) is an increasing function, \(f''\)is positive

Step-by-step solution

Find the first derivative

The given function is \(f\left( x \right) = 2x - 5{x^{3/4}}\).

Use difference rule of differentiation which implies that \(\frac{d}{{dx}}\left( {f\left( x \right) - g\left( x \right)} \right) = \frac{d}{{dx}}f\left( x \right) - \frac{d}{{dx}}g\left( x \right)\) and power rule of differentiation which implies that \(\frac{d}{{dx}}\left( {{x^n}} \right) = n{x^{n - 1}}\), where \(n\) is any real number.

On differentiating this function with respect to \(x\), the following expression is obtained:

\(\begin{aligned}f'\left( x \right) &= \frac{d}{{dx}}\left( {2x - 5{x^{3/4}}} \right)\\ &= \frac{d}{{dx}}\left( {2x} \right) - \frac{d}{{dx}}\left( {5{x^{3/4}}} \right)\\ &= 2 - \frac{{15}}{4}{x^{ - 1/4}}\end{aligned}\)

Thus, the first derivative is \(f'\left( x \right) = 2 - \frac{{15}}{4}{x^{ - 1/4}}\).

 Step 2: Find the second derivative

The first derivative of the function is \(f'\left( x \right) = 2 - \frac{{15}}{4}{x^{ - 1/4}}\). On differentiating it again, with respect to \(x\) axis, we obtain the second derivative, as follows:

\(\begin{aligned}f''\left( x \right) &= \frac{d}{{dx}}\left( {2 - \frac{{15}}{4}{x^{ - 1/4}}} \right)\\ &= \frac{d}{{dx}}\left( 2 \right) - \frac{d}{{dx}}\left( {\frac{{15}}{4}{x^{ - 1/4}}} \right)\\ &= 0 - \frac{{15}}{4}\left( { - \frac{1}{4}} \right){x^{ - 5/4}}\\ &= \frac{{15}}{{16}}{x^{ - 5/4}}\end{aligned}\)

Thus, the second derivative of function is \(f''\left( x \right) = \frac{{15}}{{16}}{x^{ - 5/4}}\).

Check the result

To check the answer visually draw the graph of the functions\(f\left( x \right) = 2x - 5{x^{3/4}}\),\(f'\left( x \right) = 2 - \frac{{15}}{4}{x^{ - 1/4}}\)and\(f''\left( x \right) = \frac{{15}}{{16}}{x^{ - 5/4}}\)by using the graphing calculator as shown below:

1. Open the graphing calculator. Select the “STAT PLOT” and enter the equation\(2x - 5{x^{3/4}}\)in the\({Y_1}\)tab,\(2 - \frac{{15}}{4}{X^{ - 1/4}}\)in \({Y_2}\)tab,\(\frac{{15}}{{16}}{X^{ - 5/4}}\) in \({Y_3}\)tab.
2. Enter the “GRAPH” button in the graphing calculator.

Visualization of graph of the functions is shown below:

Draw a conclusion

It can be noted that, as \(f\)is positive and is decreasing in nature, then \(f'\)is negative and is always increasing. As \(f'\) is an increasing function, \(f''\)is positive