Question

Given f(x) = sin^3(4x), find the rate of change of f(x) when x = π

Answer 1

To find the rate of change of a function at a point, we can take the derivative, and evaluate in that point: that value is the instantaneous rate of change of that function.

We are given the function:

`f(x)=\sin^3(4x)`

To differentiate this function, we need to use the chain rule. The chain rule tells us that if we are given two function g(x) and h(x), the derivative of their composition is:

`(g(h(x))^(\prime)=g^(\prime)(h(x))\cdot h^(\prime)(x)`

In this case, we can separate f(x) in two functions

`\begin{gathered} \begin{cases}g(x)={x^3} \\ h(x)={\sin(4x)}\end{cases} \\ f(x)=g(h(x))=\sin^3(4x) \end{gathered}`

Then, let's find the derivative:

`\begin{gathered} g(x)=x^3\Rightarrow g^(\prime)(x)=3x^2 \\ h(x)=\sin(4x)\Rightarrow h^(\prime)(x)=4\cos(4x) \end{gathered}`

By the chain rule:

`f^(\prime)(x)=3\sin^2(4x)4\cos(4x)=12\cos(4x)\sin^2(4x)`

Now, to find the value of the rate of change when x = π, we evaluate:

`f^(\prime)(\pi)=12\cos(4\pi)\sin^2(4\pi)=12\cdot1\cdot0=0`

The rate of change in x = π is 0