# Math 22 Implicit Differentiation

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## Implicit Differentiation

Consider the equation ${\displaystyle x^{2}y=5}$. To find ${\displaystyle {\frac {dy}{dx}}}$, we can rewrite the equation as ${\displaystyle y={\frac {5}{x^{2}}}}$, then differentiate as usual. ie: ${\displaystyle y={\frac {5}{x^{2}}}=5x^{-2}}$, so ${\displaystyle {\frac {dy}{dx}}=-10x^{-3}}$. This is called explicit differentiation.

However, sometimes, it is difficult to express ${\displaystyle y}$ as a function of ${\displaystyle x}$ explicitly. For example: ${\displaystyle y^{2}-2x+4xy=5}$

Therefore, we can use the procedure called implicit differentiation

## Guidelines for Implicit Differentiation

 Consider an equation involving ${\displaystyle x}$ and ${\displaystyle y}$ in which ${\displaystyle y}$ is a differentiable function of ${\displaystyle x}$. You can use the steps below to find ${\displaystyle {\frac {dy}{dx}}}$.
1. Differentiate both sides of the equation with respect to ${\displaystyle x}$.
2. Collect all terms involving ${\displaystyle {\frac {dy}{dx}}}$ on the left side of the equation and move all other terms to the right side of the equation
3. Factor ${\displaystyle {\frac {dy}{dx}}}$ out of the left side of the equation.
4. Solve for ${\displaystyle {\frac {dy}{dx}}}$ by dividing both sides of the equation by the left-hand factor that does not contain ${\displaystyle {\frac {dy}{dx}}}$.


Example: Find ${\displaystyle {\frac {dy}{dx}}}$ of

1) ${\displaystyle y^{3}=5x^{3}+8x}$

Solution:
${\displaystyle {\frac {d}{dx}}[y^{3}]={\frac {d}{dx}}[5x^{3}+8x]}$
${\displaystyle =3y^{2}{\frac {dy}{dx}}={\frac {d}{dx}}[5x^{3}]+{\frac {d}{dx}}[8x]}$
${\displaystyle =3y^{2}{\frac {dy}{dx}}=15x^{2}+8}$
${\displaystyle ={\frac {dy}{dx}}={\frac {15x^{2}+8}{3y^{2}}}}$

2) ${\displaystyle x^{2}y+2xy+7=6x}$

Solution:
${\displaystyle {\frac {d}{dx}}[x^{2}y+2xy+7]={\frac {d}{dx}}[6x]}$
${\displaystyle ={\frac {d}{dx}}[x^{2}y]+{\frac {d}{dx}}[2xy]+{\frac {d}{dx}}[7]=6}$
${\displaystyle =[2xy+x^{2}{\frac {dy}{dx}}]+[2y+2x{\frac {dy}{dx}}]+0=6}$
${\displaystyle =x^{2}{\frac {dy}{dx}}+2x{\frac {dy}{dx}}=6-2xy-2y}$
${\displaystyle =(x^{2}+2x){\frac {dy}{dx}}=6-2xy-2y}$
${\displaystyle ={\frac {dy}{dx}}={\frac {6-2xy-2y}{x^{2}+2x}}}$