Math 22 Implicit Differentiation

Implicit Differentiation

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

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

Therefore, we can use the procedure called implicit differentiation

Guidelines for Implicit Differentiation

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

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

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

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

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

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

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