Implicit Differentiation
EXPLICIT: one variable is described in terms of one other variable.
y = 2x^{3}  5
In other situations functions are described implicitly using the relationship between the two variables such as, x^{4} + y^{2} = 1  5xy.
IMPLICIT: variables are NOT is described in terms of one other variable.
x^{4} + y^{2} = 1  5xy
Functions described implicitly can be differentiated without being written explicitly. This type of differentiation is called Implicit Differentiation.
To differentiate implicitly both sides of the equation are differentiated with respect to x and the resulting equation solved for $\frac{\partial y}{\partial x}$.
Let's look at some examples:
Step 1: Differentiate the xterms on both side of the equation with respect to x. 
$\frac{\partial}{\partial x}({x}^{2}+{y}^{2})=\frac{\partial}{\partial x}36$ Apply Sum Rule. $\frac{\partial}{\partial x}{x}^{2}+\frac{\partial}{\partial x}{y}^{2}=\frac{\partial}{\partial x}36$ Apply Power and Constant Rule. $2x+\frac{\partial}{\partial x}{y}^{2}=0$ 
Step 2: Differentiate the yterms. Factor $\frac{\partial}{\partial x}$ into $\frac{\partial}{\partial y}\frac{\partial y}{\partial x}$ and take the derivative of $\frac{\partial}{\partial y}{y}^{2}$ normally resulting in the derivative times $\frac{\partial y}{\partial x}$. 
$2x+\frac{\partial}{\partial x}{y}^{2}=0$ Original $\frac{\partial}{\partial x}{y}^{2}=\frac{\partial}{\partial y}{y}^{2}\frac{\partial y}{\partial x}={2}{y}\frac{\partial y}{\partial x}$ Implicit Diff. $2x+{2}{y}\frac{\partial y}{\partial x}=0$ Substitute 
Step 3: Solve for $\frac{\partial y}{\partial x}$ and simplify. 
Original $2x+2y\frac{\partial y}{\partial x}=0$ Subtract 2x to both sides: $2y\frac{\partial y}{\partial x}=2x$ Divide both sides by 2y. $\frac{\partial y}{\partial x}=\frac{2x}{2y}$ Simplify $\frac{x}{y}$ 
Step 1: Differentiate the xterms on both side of the equation with respect to x. 
$\frac{\partial}{\partial x}(3{y}^{3}{x}^{3})=\frac{\partial}{\partial x}5xy$ Apply Difference Rule. $\frac{\partial}{\partial x}3{y}^{3}\frac{\partial}{\partial x}{x}^{3}=\frac{\partial}{\partial x}5xy$ Apply Constant Multiple Rule. $3\frac{\partial}{\partial x}{y}^{3}\frac{\partial}{\partial x}{x}^{3}=5\frac{\partial}{\partial x}xy$ Apply Power Rule. $3\frac{\partial}{\partial x}{y}^{3}3{x}^{2}=5\frac{\partial}{\partial x}xy$ 

Step 2: Differentiate the yterms. Factor $\frac{\partial}{\partial x}$ into $\frac{\partial}{\partial y}\frac{\partial y}{\partial x}$ and take the derivative of $\frac{\partial}{\partial y}$ normally resulting in the derivative times $\frac{\partial y}{\partial x}$. 
Original $3\frac{\partial}{\partial x}{y}^{3}3{x}^{2}=5\frac{\partial}{\partial x}xy$ Implicitly differentiate $3\frac{\partial}{\partial x}{y}^{3}$ using the power rule. $3\frac{\partial}{\partial x}{y}^{3}=3\frac{\partial}{\partial y}{y}^{3}\frac{\partial y}{\partial x}={9}{y}^{2}\frac{\partial y}{\partial x}$ Implicitly differentiate $5\frac{\partial}{\partial x}xy$ using the product rule. $5\frac{\partial}{\partial x}xy=5\left[\left(x\xb71\right)\frac{\partial y}{\partial x}+\left(y\xb71\right)\right]$ $5x\frac{\partial y}{\partial x}+5y$ Substitute into the original equation: ${9}{y}^{2}\frac{\partial y}{\partial x}3{x}^{2}={5}{x}\frac{\partial y}{\partial x}{+}{5}{y}$ 

Step 3: Solve for $\frac{\partial y}{\partial x}$ and simplify. 
Original $9{y}^{2}\frac{\partial y}{\partial x}3{x}^{2}=5x\frac{\partial y}{\partial x}+5y$ Add 3x^{2} to both sides: $9{y}^{2}\frac{\partial y}{\partial x}=5x\frac{\partial y}{\partial x}+5y+3{x}^{2}$ Subtract $5x\frac{\partial y}{\partial x}$ from both sides. $9{y}^{2}\frac{\partial y}{\partial x}5x\frac{\partial y}{\partial x}=5y+3{x}^{2}$ Factor out $\frac{\partial y}{\partial x}$ from the lefthand side: $\frac{\partial y}{\partial x}\left(9{y}^{2}5x\right)=5y+3{x}^{2}$ Divide both sides by 9y^{2}  5x. $\frac{\partial y}{\partial x}=\frac{5y+3{x}^{2}}{9{y}^{2}5x}$ 
Step 1: Differentiate the xterms on both side of the equation with respect to x. 
$\frac{\partial}{\partial x}\left({e}^{y}\right)=\frac{\partial}{\partial x}\left(2xy\right)$ Apply Difference Rule. $\frac{\partial}{\partial x}{e}^{y}=\frac{\partial}{\partial x}2x\frac{\partial}{\partial x}y$ Apply Constant Multiple Rule. $\frac{\partial}{\partial x}{e}^{y}=2\frac{\partial}{\partial x}x\frac{\partial}{\partial x}y$ Apply Power Rule. $\frac{\partial}{\partial x}{e}^{y}=2\frac{\partial}{\partial x}y$ 
Step 2: Differentiate the yterms. Factor $\frac{\partial}{\partial x}$ into $\frac{\partial}{\partial y}\frac{\partial y}{\partial x}$ and take the derivative of $\frac{\partial}{\partial y}$ normally resulting in the derivative times $\frac{\partial y}{\partial x}$. 
Original $\frac{\partial}{\partial x}{e}^{y}=2\frac{\partial}{\partial x}y$ Implicitly differentiate e^{y} using the natural exponent rule. $\frac{\partial}{\partial x}{e}^{y}=\frac{\partial}{\partial y}{e}^{y}\frac{\partial y}{\partial x}={{e}}^{{y}}\frac{\partial y}{\partial x}$ Implicitly differentiate $\frac{\partial}{\partial x}y$ using the power rule. $\frac{\partial}{\partial x}y=\frac{\partial}{\partial y}y\frac{\partial y}{\partial x}=1\frac{\partial y}{\partial x}$ $1\frac{\partial y}{\partial x}$ Substitute into the original equation: ${e}^{y}\frac{\partial y}{\partial x}=2{1}\frac{\partial y}{\partial x}$ 
Step 3: Solve for $\frac{\partial y}{\partial x}$ and simplify. 
Original ${e}^{y}\frac{\partial y}{\partial x}=21\frac{\partial y}{\partial x}$ Add $1\frac{\partial y}{\partial x}$ to both sides: ${e}^{y}\frac{\partial y}{\partial x}+1\frac{\partial y}{\partial x}=2$ Factor out $\frac{\partial y}{\partial x}$ from the lefthand side: $\frac{\partial y}{\partial x}\left({e}^{y}+1\right)=2$ Divide both sides by e^{y} + 1. $\frac{\partial y}{\partial x}=\frac{2}{{e}^{y}+1}$ 
Related Links: Math algebra Combination of Functions Composition of Functions Calculus Topics 
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