Problem 60
Question
Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$ f(x)=x^{3}(x-4)^{2} $$
Step-by-Step Solution
Verified Answer
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.
1Step 1: Identify the First and Second Function
First function, \(u = x^{3}\), and second function, \(v = (x-4)^{2}\).
2Step 2: 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: \(f'(x) = u'v+uv' = 3x^{2}(x-4)^{2} + x^{3} * 2 * (x-4) = 3x^{2}(x-4)^{2} + 2x^{3}(x-4)\).
3Step 3: Simplify the Result
Simplify \(f'(x) = 3x^{2}(x-4)^{2} + 2x^{3}(x-4)\) to obtain the final simplified derivative.
Other exercises in this chapter
Problem 59
Describe the \(x\) -values at which \(f\) is differentiable. $$ f(x)=\frac{1}{x-1} $$
View solution Problem 59
find the limit $$ \lim _{\Delta t \rightarrow 0} \frac{(t+\Delta t)^{2}-5(t+\Delta t)-\left(t^{2}-5 t\right)}{\Delta t} $$
View solution Problem 60
The percent \(P\) of defective parts produced by a new employee \(t\) days after the employee starts work can be modeled by \(P=\frac{t+1750}{50(t+2)}\) Find th
View solution Problem 60
The cost \(C\) (in millions of dollars) of removing \(x\) percent of the pollutants emitted from the smokestack of a factory can be modeled by \(C=\frac{2 x}{10
View solution