Derive the function x^3y*5+3y^4=xy^2*pi with respect to x

\frac{d}{dx}\left(5x^3\cdot y+3y^4=\pi x y^2\right)

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Answer

$15yx^{2}=\piy^2$

Step by step solution

Problem

$\frac{d}{dx}\left(5x^3\cdot y+3y^4=\pi x y^2\right)$
1

Apply implicit differentiation by taking the derivative of both sides of the equation with respect to the differentiation variable

$\frac{d}{dx}\left(3y^4+5yx^3\right)=\frac{d}{dx}\left(\pixy^2\right)$
2

Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=\pix$ and $g=y^2$

$\frac{d}{dx}\left(3y^4+5yx^3\right)=\pix\frac{d}{dx}\left(y^2\right)+y^2\frac{d}{dx}\left(\pix\right)$
3

The derivative of the constant function is equal to zero

$\frac{d}{dx}\left(3y^4+5yx^3\right)=0\cdot \pix+y^2\frac{d}{dx}\left(\pix\right)$
4

Any expression multiplied by $0$ is equal to $0$

$\frac{d}{dx}\left(3y^4+5yx^3\right)=0+y^2\frac{d}{dx}\left(\pix\right)$
5

The derivative of a function multiplied by a constant is equal to the constant times the derivative of the function

$\frac{d}{dx}\left(3y^4+5yx^3\right)=0+\piy^2\frac{d}{dx}\left(x\right)$
6

The derivative of the linear function is equal to $1$

$\frac{d}{dx}\left(3y^4+5yx^3\right)=0+1\cdot \piy^2$
7

The derivative of a sum of two functions is the sum of the derivatives of each function

$\frac{d}{dx}\left(3y^4\right)+\frac{d}{dx}\left(5yx^3\right)=0+1\cdot \piy^2$
8

The derivative of the constant function is equal to zero

$0+\frac{d}{dx}\left(5yx^3\right)=0+1\cdot \piy^2$
9

Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=5x^3$ and $g=y$

$0+5x^3\frac{d}{dx}\left(y\right)+y\frac{d}{dx}\left(5x^3\right)=0+1\cdot \piy^2$
10

The derivative of the constant function is equal to zero

$0+0\cdot 5x^3+y\frac{d}{dx}\left(5x^3\right)=0+1\cdot \piy^2$
11

Any expression multiplied by $0$ is equal to $0$

$0+0+y\frac{d}{dx}\left(5x^3\right)=0+1\cdot \piy^2$
12

The derivative of a function multiplied by a constant is equal to the constant times the derivative of the function

$0+0+5y\frac{d}{dx}\left(x^3\right)=0+1\cdot \piy^2$
13

The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

$0+0+5\cdot 3yx^{2}=0+1\cdot \piy^2$
14

Add the values $0$ and $0$

$15yx^{2}=0+1\cdot \piy^2$
15

Multiply $\pi$ times $1$

$15yx^{2}=0+\piy^2$
16

$x+0=x$, where $x$ is any expression

$15yx^{2}=\piy^2$

Answer

$15yx^{2}=\piy^2$

Problem Analysis

Main topic:

Differential calculus

Time to solve it:

0.22 seconds

Views:

162