Question

If f'(0) = 5 and F(x) = f(3x), what is F'(0)?

Answer 1

F'(0)=15

Explanation:

Differentiate
`F(x)` with respect to
`x`.

`F'(x)=3f'(x)`

Subsitute
`0` for
`x` in the above equation.

`F'(0)=3f'(0)\\=3\cdot5\\=15`

Answer 2

`\displaystyle F'(0) = 15`

Explanation:

We are given that:

`f'(0) = 5 \text{ and } F(x) = f(3x)`

And we want to find F'(0).

First, find F(x):

`\displaystyle F'(x) = (d)/(dx)\left[ f(3x)]`

From the chain rule:

`\displaystyle \begin{aligned} F'(x) &= f'(3x) \cdot (d)/(dx) \left[ 3x\right] \\ \\ &= 3f'(3x)\end{aligned}`

Then:

`\displaystyle \begin{aligned} F'(0) & = 3f'(3(0)) \\ \\ & = 3f'(0) \\ \\ & = 3(5) \\ \\ & = 15\end{aligned}`

In conclusion, F'(0) = 15.