"How to differentiate y = x 8 x..." solved and explained

purpuradoh89

Answered question

2023-02-11

How to differentiate

y = x^{8 x}

Miracle Hurley

Beginner2023-02-12Added 9 answers

Use logarithmic differentiation (or its equivalent exponential form).
Solution:

y = x^{8 x}

Logarithmic Differentiation
Consider the logarithms of both sides.

\ln y = \ln (x^{8 x}) = 8 x \ln x

. Now differentiate implicitly.

\frac{1}{y} \frac{d y}{d x} = 8 \ln x + 8 x (\frac{1}{x}) = 8 \ln x + 8

\frac{d y}{d x} = y (8 \ln x + 8) = x^{8 x} (8 \ln x + 8)

.
Exponential Equivalent

y = x^{8 x} = e^{\ln (x^{8 x})} = e^{8 x \ln x}

Differentiate using

\frac{d}{d x} (e^{u}) = e^{u} \frac{d u}{d x}

\frac{d y}{d x} = e^{8 x \ln x} (8 \ln x + 8)

= x^{8 x} (8 \ln x + 8)

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