If f(x) = 2x + e3x + tan(x) and g is the inverse of f(x), determine g′(1).
We will use the concept of inverse function in order to find the required answer.
Answer: If f(x) = 2x + e3x + tan(x) and g is the inverse of f(x), then g'(1) = 1/6
Let us see how we will use the concept of inverse function in order to find the required answer.
Explanation:
Since it is given that g(x) is the inverse function of f(x).
Hence, we can write as g(f(x)) = x
Now differentiating both sides with respect to x we get
g'(f(x)) f'(x) = 1
Now we are required to find g'(1).
Since we have g'(f(x)) we need to find the value of f(x) = 1, to get g'(1).
Hence, f(x) = 1. For f(x) = 1, x = 0.
f'(x) = 2 + 3e3x + sec2 (x)
f'(0) = 2 + 3 + 1 = 6.
g'(f(0)).f'(0) = 1
g'(1).6 = 1
g'(1) = 1/6
Hence, g'(1) = 1/6
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