Question

7-52: Find the derivative of the function

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

Short answer

The derivative of the function is \(f'\left( x \right) = 10x{\left( {2{x^3} - 5{x^2} + 4} \right)^4}\left( {3x - 5} \right)\).

Step-by-step solution

Power rule combined with the Chain Rule

Power rule is defined as shown below:

\(\frac{d}{{dx}}\left( {{u^n}} \right) = n{u^{n - 1}}\frac{{du}}{{dx}}\)

Additionally, \(\frac{d}{{dx}}{\left( {g\left( x \right)} \right)^n} = n{\left( {g\left( x \right)} \right)^{n - 1}} \cdot g'\left( x \right)\).

Find the derivative of the function

Use the power rule combined with the chain rule to obtain the derivative of the function as shown below:

\(\begin{aligned}f'\left( x \right) &= \frac{d}{{dx}}{\left( {2{x^3} - 5{x^2} + 4} \right)^5}\\ &= 5{\left( {2{x^3} - 5{x^2} + 4} \right)^4} \cdot \frac{d}{{dx}}\left( {2{x^3} - 5{x^2} + 4} \right)\\ &= 5{\left( {2{x^3} - 5{x^2} + 4} \right)^4}\left( {6{x^2} - 10x} \right)\\ &= 5{\left( {2{x^3} - 5{x^2} + 4} \right)^4} \cdot 2x\left( {3x - 5} \right)\\ &= 10x{\left( {2{x^3} - 5{x^2} + 4} \right)^4}\left( {3x - 5} \right)\end{aligned}\)

Thus, the derivative of the function is \(f'\left( x \right) = 10x{\left( {2{x^3} - 5{x^2} + 4} \right)^4}\left( {3x - 5} \right)\).