Question

If \(f\left( x \right) = {\rm{sin}}x + {\rm{ln}}x\), find \(f'\left( x \right)\).Check that your answer is reasonable by comparing the graphs of \(f\) and \(f'\).

Short answer

The required answer is \(f'\left( x \right) = \cos x + \frac{1}{x}\).

The graph of the function\(f\left( x \right) = \sin x + \ln x\),and \(f'\left( x \right) = \cos x + \frac{1}{x}\) is shown below:

On observing the above graph, it can be concluded that this is reasonable because the graph shows that as the function \(f\) increases when the function \(f'\) is positive.

If the function \(f\) has a horizontal tangent, then, \(f'\left( x \right) = 0\).

Step-by-step solution

Write the formula of the derivatives of logarithmic functions, the chain rule and the power rule.

Derivative of logarithm function: \(\frac{d}{{dx}}\left( {\ln x} \right) = \frac{1}{x}\),

Theequation of the tangent lineto the curve at the given point \(\left( {{x_1},{y_1}} \right)\) is \(y - {y_1} = \frac{{dy}}{{dx}}\left( {x - {x_1}} \right)\).

Find the differentiation of the function

Consider the function\(f\left( x \right) = \sin x + \ln x\). Differentiate the function w.r.t \(x\) by using the derivatives of logarithmic functions.

\(\begin{aligned}{c}\frac{d}{{dx}}\left( {f\left( x \right)} \right)&=frac{d}{{dx}}\left( {\sin x + \ln x} \right)\\&= \frac{d}{{dx}}\left( {\sin x} \right) + \frac{d}{{dx}}\left( {{\rm{ln}}x} \right)\\ &= \cos x + \frac{1}{x}\end{aligned}\)

Step 3: Compare the graphs of \(f\) and \(f'\).

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\({f_1}\left( x \right) = \sin x + \ln x\),and \({f_2}\left( x \right) = \cos x + \frac{1}{x}\)by using the graphing calculator as shown below:

1. Open the graphing calculator. Select the “STAT PLOT” and enter the equation\(\sin x + \ln x\)in the\({Y_1}\)tab.
2. Select the “STAT PLOT” and enter the equation\(\cos x + \frac{1}{x}\)in the\({Y_2}\)tab.
3. Enter the “GRAPH” button in the graphing calculator.

Visualization of graph of the function\({f_1}\left( x \right) = \sin x + \ln x\),and \({f_2}\left( x \right) = \cos x + \frac{1}{x}\) is shown below:

On observing the above graph, it can be concluded that this is reasonable because the graph shows that as the function \(f\) increases when the function \(f'\) is positive.

If the function \(f\) has a horizontal tangent, then,\(f'\left( x \right) = 0\).