Find f'(x) for f(x)=(2x^2+5)^7
A. 7(4x)^6
B. 7(4x)^7
C. 28x(2x^2+5)^7
D. 7(2x^2+5)^6
E. None of these
Would it be none of these because of the 4x in the chain rule?
To find the derivative of the function f(x)=(2x^2+5)^7, we can use the chain rule.
The chain rule states that if we have a composite function g(f(x)), then the derivative of g(f(x)) with respect to x is given by g'(f(x)) * f'(x).
In this case, the outer function is g(u) = u^7, where u = 2x^2 + 5. So, we need to find the derivative of the outer function g(u) and the derivative of the inner function u = 2x^2 + 5.
To find the derivative of the outer function g(u) = u^7, we can use the power rule. The power rule states that if we have a function h(x) = x^n, then its derivative is given by h'(x) = n * x^(n-1).
So, the derivative of the outer function g(u) = u^7 is g'(u) = 7u^(7-1) = 7u^6.
Now, let's find the derivative of the inner function u = 2x^2 + 5. Since u is a function of x, we need to apply the power rule and the chain rule.
Taking the derivative of u = 2x^2 + 5, we get u' = 2(2x) = 4x.
Now, using the chain rule, we can find the derivative of f(x)=(2x^2+5)^7.
f'(x) = g'(u) * u'
= 7u^6 * 4x
= 28x(2x^2 + 5)^6.
Therefore, the correct answer is C. 28x(2x^2+5)^7.