Implicit Differentiation:
Implicit functions: If [katex]y[/katex] and [katex]x[/katex] are mixed up and [katex]y[/katex] cannot be expressed in terms of the independent variable [katex]x[/katex], Then [katex]y[/katex] is called an Implicit functions. Symbolically it is written as [katex]f\left(x,y\right)=0[/katex]
Examples:
[katex]x^3+xy-x=7[/katex]
[katex]y^3+xy-x=0[/katex]
[katex]y^3+xy-xy^3=0[/katex]
Procedure:
- Step (1) when [katex]x[/katex] and [katex]y[/katex] are related Implicit we assumethat [katex]y[/katex] is differentiable function of [katex]x[/katex]
- Step (2) Differentiate both sides of eq w.r.t [katex]x[/katex]
- Step (3) Solve the resulting eq for [katex]\frac{dy}{dx}[/katex]
Example 1: Implicit Differentiation
[katex]x^3+xy-x=7[/katex]
Differentiate w.r.t [katex]x[/katex]
[katex]\frac d{dx}\left(x^3+xy-x\right)=\frac d{dx}7[/katex]
[katex]\frac d{dx}\left(x^3\right)+\frac d{dx}\left(xy\right)-\frac d{dx}(x)=\frac d{dx}7[/katex] [sum and difference rule]
[katex]3x^2+\left[y\frac d{dx}x+x\frac d{dx}y\right]-1=0[/katex]
[katex]3x^2+y+x\frac d{dx}y=1[/katex]
[katex]x\frac d{dx}y=1-3x^2-y[/katex]
[katex]\boxed{\frac d{dx}y=\frac{1-3x^2-y}x}[/katex]
Example 2: [implicit Differentiation]
[katex]y^3+xy-x=0[/katex]
Differentiate w.r.t [katex]x[/katex]
[katex]\frac d{dx}\left(y^3+xy-x\right)=\frac d{dx}0[/katex]
[katex]\frac d{dx}y^3+\frac d{dx}xy-\frac d{dx}x=\frac d{dx}0[/katex] [sum and difference rule]
[katex]3y^2+y\frac d{dx}x+x\frac d{dx}y-1=0[/katex]
[katex]3y^2+y+x\frac d{dx}y=1[/katex]
[katex]x\frac d{dx}y=1-3y^2-y[/katex]
[katex]\boxed{\frac d{dx}y=\frac{1-3y^2-y}x}[/katex]
Example 3: [implicit Differentiation]
[katex]y^3+y-xy^3=0[/katex]
Differentiate w.r.t [katex]x[/katex]
[katex]\frac d{dx}\left(y^3+y-xy^3\right)=\frac d{dx}0[/katex]
[katex]\frac d{dx}y^3+\frac d{dx}y-\frac d{dx}xy^3=0[/katex] [sum and difference rule]
[katex]3y^2+\frac d{dx}y-\left[y^3\frac d{dx}x+x\frac d{dx}y^3\right]=0[/katex]
[katex]3y^2+\frac d{dx}y-\left[y^3+3xy^2\frac{dy}{dx}\right]=0[/katex]
[katex]3y^2+\frac d{dx}y-y^3-3xy^2\frac{dy}{dx}=0[/katex]
[katex]\frac d{dx}y-3xy^2\frac{dy}{dx}=y^3-3y^2[/katex]
[katex]\frac d{dx}\left(y-3xy^2\right)=y^3-3y^2[/katex]
[katex]\frac{dy}{dx}=\frac{y^3-3y^2}{y-3xy^2}[/katex]
[katex]\frac{dy}{dx}=\frac{y\left(y^2-3y\right)}{y(1-3xy)}[/katex]
[katex]\boxed{\frac{dy}{dx}=\frac{y^2-3y}{1-3xy}}[/katex]