Key to "Find the derivatives of the functions. u(x)=3x2x2+1"

High School
Calculus and Analysis
Calculus 1
Derivatives

Lipossig

Answered question

2021-06-23

Find the derivatives of the functions.

u (x) = \frac{3^{x^{2}}}{x^{2} + 1}

Answer & Explanation

Layton

Skilled2021-06-24Added 89 answers

$u (x) = \frac{3^{x^{2}}}{x^{2} + 1}$ Differentiate both sides with respect to x
$u^{'} (x) = \frac{d}{d x} [\frac{3^{x^{2}}}{x^{2} + 1}]$
Apply the Quotient Rule
$u^{'} (x) = (x^{2} + 1) \frac{d}{d x} [3^{x^{2}}] - 3 x^{2} \frac{d}{d x} [x^{2} + 1]$
Apply $\frac{d}{d x} [b^{u}] = b^{u} \ln b \frac{d u}{d x}$ and $\frac{d}{d x} [x^{n}] = n x^{n - 1}$
$u^{'} (x) = \frac{(x^{2} + 1) (3^{x^{2}}) (\ln 3) \frac{d}{d x} [x^{2}] - 3^{x^{2}} (2 x^{2 - 1})}{(x^{2} + 1)^{2}}$
Simplify
$u^{'} (x) = \frac{(x^{2} + 1) (3^{x^{2}}) (\ln 3) (2 x) - 3^{x^{2}} (2 x)}{{(x^{2} + 1)}^{2}}$
Factor out $2 x \cdot 3^{x^{2}}$
$u^{'} (x) = \frac{2 x \cdot 3^{x^{2}} [(x^{2} + 1) \ln 3 - x]}{{(x^{2} + 1)}^{2}}$

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