# Find the differential dy when y=rad(15+x^2).Evaluate dy for the given values of x and dx. x = 1, dx = −0.2

This article aims to find the differential of a given equation and the value of differential for given values of other parameters. Readers should know about differential equations and their basics to solve problems like in this article.

A differential equation is defined as an equation contains one or more terms and the derivatives of one variable (i.e., the dependent variable) concerning another variable (i.e., the independent variable)

$\dfrac{dy}{dx} = f(x)$

$x$ reprsents an independent variable, and $y$ is dependent variable.

Given

$y = \sqrt { 15 + x ^ { 2 } }$

The differential of $y$ is the derivative of a function times the differential of $x$.

Therefore,

$dy = \dfrac { 1 } { 2 \sqrt { 15 + x ^ { 2 } } } . \dfrac { d } { dx } ( 15 + x ^ { 2 } ) . dx$

$\Rightarrow dy = \dfrac{1}{2 \sqrt {15+x^{2}}}.(0+2x)dx$

$dy = \dfrac{x}{\sqrt {15+x^{2}}} dx$

Part (b)

Substituting $x= 1$ and $dx = -0.2$ in $dy$ , we get

$\Rightarrow dy = \dfrac { 1 } { 15 + ( 1 ) ^ { 2 } } ( – 0.2 )$

$\Rightarrow dy = \dfrac { 1 } { \sqrt { 16 } } (- 0.2 )$

$\Rightarrow dy = \dfrac { – 0.2 } { 4 }$

$\Rightarrow dy = – 0.05$

The value of $dy$ for $x= 1$ and $dx = -0.2$ is $-0.05$

## Numerical Result

– The differential $dy$ is given as:

$dy = \dfrac { x } { \sqrt { 15 + x ^ { 2 }}} dx$

– The value of $dy$ for $x= 1$ and $dx = -0.2$ is $-0.05$

## Example

(a) Find the differential $dy$ for $y = \sqrt { 20 – x ^ { 3 }}$.

(b) Evaluate $dy$ for given values of $x$ and $dx$. $x = 2$, $dx = – 0.2$.

Solution

Given

$y = \sqrt { 20 – x ^ { 3 } }$

The differential of $y$ is the derivative of a function times the differential of $x$.

Therefore,

$dy = \dfrac {1} {2\sqrt { 20 – x^{3}}}.\dfrac { d } { dx } (20-x^{3}).dx$

$\Rightarrow dy = \dfrac{1}{2 \sqrt {20-x^{3}}}.(0-3x^{2})dx$

$dy = \dfrac{-3x^{2}}{2\sqrt {20-x^{3}}} dx$

Part (b)

Substituting $x= 2$ and $dx = -0.2$ in $dy$ , we get

$\Rightarrow dy = \dfrac {-3( 2 ) ^ { 2 } } { 2\sqrt {20 – (2) ^ { 3 }}} (- 0.2)$

$\Rightarrow dy = \dfrac { -12 } { 4\sqrt { 3 }}(- 0.2)$

$\Rightarrow dy = \dfrac { 2.4 } { 4 \sqrt { 3 } }$

$\Rightarrow dy = 0.346$

The value of $dy$ for $x= 2$ and $dx = -0.2$ is $0.346$