Question

Write the composite function in the form \(f\left( {g\left( x \right)} \right)\). (Identify the inner function \(u = g\left( x \right)\) and the outer function \(y = f\left( u \right)\).) Then find the derivative \(\frac{{dy}}{{dx}}\).

6. \(y = \sqrt(3){{{e^x} + 1}}\)

Short answer

The inner function and outer function are \(u = g\left( x \right) = {e^x} + 1\) and \(y = f\left( u \right) = {u^{\frac{1}{3}}}\). The derivative of the function is \(\frac{{dy}}{{dx}} = \frac{{{e^x}}}{{3\sqrt(3){{{{\left( {{e^x} + 1} \right)}^2}}}}}\).

Step-by-step solution

The Chain Rule

The composite function \(F = f \circ g\)is denoted by \(F\left( x \right) = f\left( {g\left( x \right)} \right)\). The differentiation of this function is obtained by using chain rule as shown below:

\(F'\left( x \right) = f'\left( {g\left( x \right)} \right) \cdot g'\left( x \right)\)

By using Leibniz notation,it is defined as:

\(\frac{{dy}}{{dx}} = \frac{{dy}}{{du}}\frac{{du}}{{dx}}\)

Identify the inner function and outer function

Let \(u = g\left( x \right) = {e^x} + 1\)be the inner function.

Consider the outer function as shown below:

\(\begin{aligned}y &= f\left( u \right)\\ &= \sqrt(3){u}\\ &= {u^{\frac{1}{3}}}\end{aligned}\)

Differentiate the function

Use Leibniz notation to differentiate the function as shown below:

\(\begin{aligned}\frac{{dy}}{{dx}} &= \frac{{dy}}{{du}} \cdot \frac{{du}}{{dx}}\\ &= \frac{d}{{du}}\left( {{u^{\frac{1}{3}}}} \right) \cdot \frac{d}{{dx}}\left( {{e^x} + 1} \right)\\ &= \frac{1}{3}{u^{\frac{{ - 2}}{3}}} \cdot {e^x}\\ &= \frac{1}{{3\sqrt(3){{{{\left( {{e^x} + 1} \right)}^2}}}}} \cdot {e^x}\\ &= \frac{{{e^x}}}{{3\sqrt(3){{{{\left( {{e^x} + 1} \right)}^2}}}}}\end{aligned}\)

Thus, the derivative of the function is \(\frac{{dy}}{{dx}} = \frac{{{e^x}}}{{3\sqrt(3){{{{\left( {{e^x} + 1} \right)}^2}}}}}\).