Find out the answer to "Derivatives Find the derivative of the following functions..." - Plainmath

sputneyoh

Answered question

2021-12-07

Derivatives Find the derivative of the following functions for the derivative of

\sqrt{x} . f (x) = 3 e^{x} + 5 x + 5

Answer & Explanation

encolatgehu

Beginner2021-12-08Added 27 answers

Step 1
Given:

\sqrt{x} f (x) = 3 e^{x} + 5 x + 5

(Let us denote f(x) by 'y')
Formula: (uv)'=uv'+vu' (Product rule of differentiation)
Therefore,

\sqrt{x} y = 3 e^{x} + 5 x + 5

...(2)
Step 2
from (2) we differentiate both sie with respect to x

\sqrt{x} y^{'} + y \frac{1}{2 \sqrt{x}} = 3 e^{x} + 5

\Rightarrow \sqrt{x} y^{'} + \frac{3 e^{x} + 5 x + 5}{2 x} = 3 e^{x} + 5

\Rightarrow 2 x \sqrt{x} y^{'} + 3 e^{x} + 5 x + 5 = 6 x e^{x} + 10 x

\Rightarrow y^{'} = \frac{3 e^{x} (2 x - 1) + 5 x - 5}{2 x \sqrt{x}}

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