Question: "Evaluate the following derivatives. ddx[(2x)4x]" - Plainmath

sanuluy

Answered question

2021-01-19

Evaluate the following derivatives.

\frac{d}{d x} [{(2 x)}^{4 x}]

Answer & Explanation

lamusesamuset

Skilled2021-01-20Added 93 answers

Step 1
Given,

\frac{d}{d x} [{(2 x)}^{4 x}]

Step 2
Given equation can be written as

\frac{d}{d x} [{(2 x)}^{4 x}] = 2 \frac{d}{d x} [{(x)}^{4 x}]

We can write

x^{4 x}

as follows

x^{4 x} = e^{\ln (x^{4 x})}

e^{4 x \ln (x)}

Hence

\frac{d}{d x} [{(2 x)}^{4 x}] = 2 \frac{d}{d x} [e^{4 x \ln (x)}]

We apply the chain rule

\frac{d}{d x} [{(2 x)}^{4 x}] = 2 [e^{4 x \ln (x)}] \frac{d}{d x} [4 x \ln (x)]

= 2 [e^{4 x \ln (x)}] 4 [x \times \frac{1}{x} + \ln (x)]

= 2 [e^{4 x \ln (x)}] [1 + \ln (x)]

2 x^{4 x} [1 + l (x)]

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