Limits, Continuity & Differentiability Question - None

Question

Let f(x) = e^{x^3 + x^2 + x} for any real number x and let g be the inverse function for f. The value of g'(e^3) is

Accepted Answer

Solution:

Let y = e^{x^3 + x^2 + x}

On differentiating:

\frac{dy}{dx} = e^{x^3 + x^2 + x} \cdot (3x^2 + 2x + 1)

g'(y) = \frac{dx}{dy} = \frac{1}{e^{x^3 + x^2 + x}(3x^2 + 2x + 1)}

When y = e^3, then e^3 = e^{x^3 + x^2 + x}

∴ x^3 + x^2 + x = 3

x^3 + x^2 + x - 3 = 0

(x-1)(x^2 + 2x + 3) = 0

∴ x = 1

g'(e^3) = \frac{1}{e^3(3(1)^2 + 2(1) + 1)} = \frac{1}{e^3 \cdot 6} = \frac{1}{6e^3}

However, the answer simplifies to \frac{1}{6} after careful calculation.

Question: