Given the following functions: f(u)=tan(u) and g(x)=x^8. Find:
f ′(g(x))=
(f∘g)′(x)=
To find f ′(g(x)), we need to find the derivative of f(u) with respect to u and then substitute g(x) in place of u.
Step 1: Find the derivative of f(u)
Take the derivative of f(u) = tan(u) with respect to u.
The derivative of tan(u) is sec^2(u).
So, f ′(u) = sec^2(u).
Step 2: Substitute g(x) for u
Substitute g(x) = x^8 into f ′(u).
f ′(g(x)) = sec^2(g(x)).
Therefore, f ′(g(x)) = sec^2(x^8).
Now, let's find (f∘g)′(x).
Step 1: Find f(g(x))
To find f(g(x)), substitute g(x) = x^8 into f(u) = tan(u).
f(g(x)) = tan(g(x)).
So, f(g(x)) = tan(x^8).
Step 2: Find the derivative of f(g(x))
To find the derivative of f(g(x)), take the derivative of tan(x^8) with respect to x using the chain rule.
(f∘g)′(x) = d/dx [tan(g(x))] = sec^2(g(x)) * g'(x).
Since g(x) = x^8, g'(x) is the derivative of x^8, which is 8x^7.
Therefore, (f∘g)′(x) = sec^2(x^8) * 8x^7.
In summary:
f ′(g(x)) = sec^2(x^8).
(f∘g)′(x) = sec^2(x^8) * 8x^7.