Find the derivative of x^3x^3y^2*-1=3pi^0.5

\frac{d}{dx}\left(x^3-x^3 y^2=3\sqrt{\pi }\right)

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Answer

$3x^{2}+0-3y^2x^{2}=0$

Step by step solution

Problem

$\frac{d}{dx}\left(x^3-x^3 y^2=3\sqrt{\pi }\right)$
1

Multiply $3$ times $\sqrt{3}$

$\frac{d}{dx}\left(x^3-y^2x^3=\sqrt{28}\right)$
2

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

$\frac{d}{dx}\left(x^3-y^2x^3\right)=\frac{d}{dx}\left(\sqrt{28}\right)$
3

The derivative of the constant function is equal to zero

$\frac{d}{dx}\left(x^3-y^2x^3\right)=0$
4

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

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

Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=-x^3$ and $g=y^2$

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

The derivative of the constant function is equal to zero

$\frac{d}{dx}\left(x^3\right)+0\left(-1\right)x^3+y^2\frac{d}{dx}\left(-x^3\right)=0$
7

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

$\frac{d}{dx}\left(x^3\right)+0+y^2\frac{d}{dx}\left(-x^3\right)=0$
8

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

$\frac{d}{dx}\left(x^3\right)+0-y^2\frac{d}{dx}\left(x^3\right)=0$
9

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

$3x^{\left(3-1\right)}+0-1\cdot 3y^2x^{2}=0$
10

Subtract the values $3$ and $-1$

$3x^{2}+0-1\cdot 3y^2x^{2}=0$
11

Multiply $3$ times $-1$

$3x^{2}+0-3y^2x^{2}=0$

Answer

$3x^{2}+0-3y^2x^{2}=0$

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Problem Analysis

Main topic:

Differential calculus

Time to solve it:

0.24 seconds

Views:

83