Find the derivative of -2y+3x^3+y^3=4x+1

\frac{d}{dx}\left(y^3-2y+3x^3=4x+1\right)

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Answer

$9x^{2}=4$

Step by step solution

Problem

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

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

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

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

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

The derivative of the constant function is equal to zero

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

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

$3\frac{d}{dx}\left(x^3\right)+0+0=0+4\frac{d}{dx}\left(x\right)$
5

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

$3\frac{d}{dx}\left(x^3\right)+0+0=0+1\cdot 4$
6

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

$3\cdot 3x^{2}+0+0=0+1\cdot 4$
7

Add the values $0$ and $0$

$9x^{2}=0+1\cdot 4$
8

Multiply $4$ times $1$

$9x^{2}=0+4$
9

Add the values $4$ and $0$

$9x^{2}=4$

Answer

$9x^{2}=4$

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Problem Analysis

Main topic:

Differential calculus

Time to solve it:

0.2 seconds

Views:

86