Question

Find \(f'\left( x \right)\). Check your answer is reasonable by comparing the graphs of \(f\), and \(f'\).

79. \(f\left( x \right) = \sqrt {1 - {x^2}} \arcsin x\)

Short answer

The graph of the curve of \(f'\) crosses the \(x\)-axis where the graph of \(f\) has a horizontal tangent. The graph of \(f\) is decreasing where \(f'\) is negative and \(f\) is increasing where \(f'\) is positive.

Step-by-step solution

The product rule of differentiation

If a function\(h\left( x \right) = f\left( x \right) \cdot g\left( x \right)\)and\(f\),\(g\)are both differentiable, then the derivative of\(h\left( x \right)\)is as follows:

\(\frac{{dh}}{{dx}} = \frac{d}{{dx}}\left( {f\left( x \right) \cdot g\left( x \right)} \right) = f\left( x \right) \cdot \frac{d}{{dx}}\left( {g\left( x \right)} \right) + g\left( x \right) \cdot \frac{d}{{dx}}\left( {f\left( x \right)} \right)\)

The derivative of the function

Differentiate \(f\left( x \right)\) with respect to \(x\) as follows:

\(\begin{aligned}f'\left( x \right) &= \frac{d}{{dx}}\left( {\sqrt {1 - {x^2}} \arcsin x} \right)\\ &= \arcsin x\frac{d}{{dx}}\left( {\sqrt {1 - {x^2}} } \right) + \sqrt {1 - {x^2}} \frac{d}{{dx}}\left( {\arcsin x} \right)\\ &= \arcsin x\left( {\frac{1}{{2\sqrt {1 - {x^2}} }} \cdot \frac{d}{{dx}}\left( {1 - {x^2}} \right)} \right) + \sqrt {1 - {x^2}} \cdot \left( {\frac{1}{{\sqrt {1 - {x^2}} }}} \right)\\ &= \arcsin x\left( {\frac{1}{{2\sqrt {1 - {x^2}} }} \cdot \left( { - 2x} \right)} \right) + 1\\ &= - \frac{{x\arcsin x}}{{\sqrt {1 - {x^2}} }} + 1\end{aligned}\)

Hence, \(f'\left( x \right) = 1 - \frac{{x\arcsin x}}{{\sqrt {1 - {x^2}} }}\).

Check the answer visually

The procedure to draw the graph of the above equation by using the graphing calculator is as follows:

To check the answer, visually draw the graph of the function\(f\left( x \right) = \sqrt {1 - {x^2}} \arcsin x\), \(f'\left( x \right) = 1 - \frac{{x\arcsin x}}{{\sqrt {1 - {x^2}} }}\) using the graphing calculator as shown below:

1. Open the graphing calculator. Select the “STAT PLOT” and enter the equation\(\sqrt {1 - {x^2}} \arcsin x\)in the\({Y_1}\)tab.
2. Enter the equation\(1 - \frac{{x\arcsin x}}{{\sqrt {1 - {x^2}} }}\)in the\({Y_2}\)tab.
3. Enter the “GRAPH” button in the graphing calculator.

Visualization of the graph of the function\(f\left( x \right) = \sqrt {1 - {x^2}} \arcsin x\), \(f'\left( x \right) = 1 - \frac{{x\arcsin x}}{{\sqrt {1 - {x^2}} }}\) is shown below:

It is clear from the graph that the curve of \(f'\) crosses the \(x\)-axis where the graph of \(f\) has a horizontal tangent.

The graph of \(f\) is decreasing where \(f'\) is negative and \(f\) is increasing where \(f'\) is positive.