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Chapter 7: Problem 60

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$ f(x)=x^{3}(x-4)^{2} $$

Short Answer

Expert verified
The derivative of \(f(x)=x^{3}(x-4)^{2}\) is \(f'(x) = 3x^{2}(x-4)^{2} + 2x^{3}(x-4)\). The Product Rule was used to find the derivative.

Step by step solution

01

Identify the First and Second Function

First function, \(u = x^{3}\), and second function, \(v = (x-4)^{2}\).
02

Apply the Product Rule

The Product Rule is \(\(uv' + u'v\)\). To apply it, first find the derivatives of \(u\) and \(v\). The derivative of \(u = x^{3}\) is \(u' = 3x^{2}\) and the derivative of \(v = (x-4)^{2}\) is \(v' = 2(x-4)\). Using these derivatives and applying the product rule will result in: \(f'(x) = u'v+uv' = 3x^{2}(x-4)^{2} + x^{3} * 2 * (x-4) = 3x^{2}(x-4)^{2} + 2x^{3}(x-4)\).
03

Simplify the Result

Simplify \(f'(x) = 3x^{2}(x-4)^{2} + 2x^{3}(x-4)\) to obtain the final simplified derivative.

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Most popular questions from this chapter

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$ y=\left(\frac{4 x^{2}}{3-x}\right)^{3} $$

Use a symbolic differentiation utility to find the derivative of the function. Graph the function and its derivative in the same viewing window. Describe the behavior of the function when the derivative is zero. $$ f(x)=\frac{\sqrt{x}+1}{x^{2}+1} $$

Use the given information to find \(f^{\prime}(2)\) \(g(2)=3\) and \(g^{\prime}(2)=-2\) \(h(2)=-1 \quad\) and \(\quad h^{\prime}(2)=4\) $$ f(x)=\frac{g(x)}{h(x)} $$

Use the given information to find \(f^{\prime}(2)\) \(g(2)=3\) and \(g^{\prime}(2)=-2\) \(h(2)=-1 \quad\) and \(\quad h^{\prime}(2)=4\) $$ f(x)=g(x)+h(x) $$

Use the General Power Rule to find the derivative of the function. $$ y=2 \sqrt{4-x^{2}} $$
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