Consider a function f(x) = x^2 + e^x then f'(f(0)) is equal to
f ' (x) = 2x + e^x
since f(0) = 0 + e^0 = 1
f'(f(0)) = f'(1) = 2 + e
Thanks, I got the same
To find f'(f(0)), we will start by calculating f(0).
Given f(x) = x^2 + e^x, we substitute x = 0 into the function:
f(0) = 0^2 + e^0
= 0 + 1
= 1
So, f(0) is equal to 1.
Next, we will find f'(x), which is the derivative of f(x).
To find the derivative of x^2, we use the power rule, which states that the derivative of x^n is n*x^(n-1):
f'(x) = d/dx (x^2) + d/dx (e^x)
= 2*x^(2-1) + e^x
= 2*x + e^x
Now, we can calculate f'(f(0)), which means plugging in f(0) into f'(x):
f'(f(0)) = f'(1)
Substituting x = 1 into f'(x):
f'(1) = 2*1 + e^1
= 2 + e
Therefore, f'(f(0)) is equal to 2 + e.
To find the value of f'(f(0)), we need to find the derivative of the function f(x) with respect to x, and then evaluate it at the point f(0).
Let's break down the process step by step:
Step 1: Find the derivative of f(x):
To find the derivative of f(x), we need to differentiate each term separately. The derivative of x^2 with respect to x is 2x, and the derivative of e^x with respect to x is e^x. Therefore, the derivative of f(x) is given by f'(x) = 2x + e^x.
Step 2: Evaluate f'(x) at f(0):
To evaluate f'(x) at f(0), we substitute x = f(0) into the derivative. Since f(0) = 0^2 + e^0 = 0 + 1 = 1, we substitute x = 1 into f'(x). Therefore, f'(f(0)) is equal to f'(1).
Step 3: Substitute x = 1 into f'(x):
Substituting x = 1 into f'(x) = 2x + e^x, we get f'(1) = 2(1) + e^1 = 2 + e.
So, f'(f(0)) is equal to 2 + e.