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 ${\frac {dy}{dx}}$.

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