If f(x) = 2x + e^3x + 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 + e^3x + 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 + 3e^3x + 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