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 \(f''\).

52. \(f\left( x \right) = {e^x} - {x^3}\)

Short answer

The first derivative is \(f'\left( x \right) = {e^x} - 3{x^2}\)and the second derivative is \(f'\left( x \right) = {e^x} - 6x\).

From the graph, as curve of\(f\)move horizontally, then \(f'\)is 0 and as curve of\(f'\)moves horizontally, then \(f''\)is 0.

Step-by-step solution

Find the first derivative

The given function is \(f\left( x \right) = {e^x} - {x^3}\).

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:

\(\begin{aligned}f'\left( x \right) &= \frac{d}{{dx}}\left( {{e^x} - {x^3}} \right)\\ &= \frac{d}{{dx}}\left( {{e^x}} \right) - \frac{d}{{dx}}\left( {{x^3}} \right)\\ &= {e^x} - 3{x^2}\end{aligned}\)

Thus, the first derivative is \(f'\left( x \right) = {e^x} - 3{x^2}\).

Find the second derivative

The first derivative of the function is \(f'\left( x \right) = {e^x} - 3{x^2}\). On differentiating it again, with respect to \(x\) axis, we obtain the second derivative, as follows:

\(\begin{aligned}f''\left( x \right) &= \frac{d}{{dx}}\left( {{e^x} - 3{x^2}} \right)\\ &= \frac{d}{{dx}}\left( {{e^x}} \right) - \frac{d}{{dx}}\left( {3{x^2}} \right)\\ &= {e^x} - 6x\end{aligned}\)

Thus, the second derivative is \(f''\left( x \right) = {e^x} - 6x\).

Check the result

To check the answer visually draw the graph of the functions\(f\left( x \right) = {e^x} - {x^3}\),\(f'\left( x \right) = {e^x} - 3{x^2}\)and\(f''\left( x \right) = {e^x} - 6x\)by using the graphing calculator as shown below:

1. Open the graphing calculator. Select the “STAT PLOT” and enter the equation\({e^X} - {X^3}\)in the\({Y_1}\)tab,\({e^X} - 3{X^2}\)in \({Y_2}\)tab,\({e^X} - 6X\) 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 the curve of\(f\)moves horizontally, then \(f'\)is 0, and as the curve \(f'\)moves horizontally, then \(f''\)is 0.