Evaluate f'(1) if f(x) =(7x+1)^2
To evaluate f'(1) for f(x) = (7x + 1)^2, we need to find the derivative of f(x) with respect to x and then substitute x = 1 into the derivative expression.
Let's find the derivative of f(x) using the chain rule. The chain rule states that if we have a function in the form (g(x))^n, then the derivative of that function is n * (g(x))^(n-1) * g'(x), where g'(x) is the derivative of the inner function g(x).
In this case, g(x) = 7x + 1 and n = 2. So, let's find g'(x) first.
The derivative of g(x) = 7x + 1 can be found by applying the power rule. The power rule states that the derivative of x^n, where n is any real number, is n * x^(n-1).
Using the power rule, we can find g'(x) as follows:
g'(x) = 7
Now, let's apply the chain rule to find the derivative of f(x):
f'(x) = 2 * (7x + 1)^(2 - 1) * g'(x)
= 2 * (7x + 1) * g'(x)
= 2 * (7x + 1) * 7
= 14 * (7x + 1)
Finally, we substitute x = 1 into f'(x) to find f'(1):
f'(1) = 14 * (7(1) + 1)
= 14 * (7 + 1)
= 14 * 8
= 112
Therefore, f'(1) is equal to 112.