"Derivatives Find the derivative of the following functions..." solved and explained

inlays85k5

Answered question

2021-11-18

Derivatives Find the derivative of the following functions

y = e^{x} x^{e}

Ralph Lester

Beginner2021-11-19Added 16 answers

Step 1
To Determine:
Derivatives Find the derivative of the following functions
Given: we have

y = e^{x} x^{e}

.
Explanation: we have a function

y = e^{x} x^{e}

by the product rule we have

\frac{d}{d x} (f \cdot g) = f \frac{d g}{d x} + g \frac{d f}{d x}

using product rule we have

f = e^{x}, g = x^{e}

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

and we know that

\frac{d e^{x}}{d x} = e^{x} and \frac{{d x}^{e}}{d x} = e x^{e - 1}

now putting these value in the above differentiation then we have
Step 2

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

= e^{x} (e x^{e - 1}) + x^{e} (e^{x})

= e x^{e - 1} e^{x} + e^{x} x^{e}

now simplifying the expression using exponent rule.

e e^{x} = e^{1 + x}

powers will be added when base is same so we have

y^{'} (x) = e^{1 + x} x^{e - 1} + e^{x} x^{e}

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