If F(x) = f( 5f( 3f( x ))), where f(0) = 0 and f '(0) = 3, find F '(0).
Well, well, well, look at this mathematical riddle! To find F '(0), we need to differentiate the function F(x) step by step. Hold on to your funny bones, here we go!
First, let's use the chain rule to differentiate F(x) from the inside out. We start with the derivative of the innermost function 3f(x), which we'll call g(x).
g(x) = 3f(x)
Next, we differentiate the function inside g(x), which is f(3f(x)). Let's call this h(x).
h(x) = f(3f(x))
Now, we differentiate the function inside h(x), which is 5f(3f(x)). Let's call this j(x).
j(x) = 5f(3f(x))
Finally, we differentiate the function inside j(x), which is F(x).
F(x) = f(j(x))
Now, let's put it all together, starting from the innermost derivative.
h'(x) = f'(3f(x)) * 3f'(x)
Next, we move to the derivative of the function j(x).
j'(x) = f'(j(x)) * h'(x)
Finally, we differentiate F(x) using the chain rule once more.
F'(x) = f'(j(x)) * j'(x)
Now, all we have to do is evaluate F'(0), according to the given conditions.
Since f(0) = 0, we know that j(0) = 5f(3f(0)) = 5f(0) = 5 * 0 = 0.
Using the condition f'(0) = 3 and plugging in j(0), we find:
F'(0) = f'(0) * j'(0) = 3 * f'(j(0)) = 3 * f'(0) = 3 * 3 = 9.
So, the answer is F'(0) = 9!
Haha, that wasn't too tricky, was it? Take this answer with a side of laughter and enjoy your day!