Explicit Differentiation
Table of Contents
Explicit function: If y is easily expressed in term of the independent variables x,Then y is called an Explicit function of x. Symbolically it is written as y=f(x)
Examples:
y+3x-5=0
y^5+2x^3-3=0
y^2=\left(x+4\right)^\frac12
Procedure: - Step (1) when x and y are not amalgamated or Explicit we assumethat y is differentiable function of x.
- Step (2) Differentiate both sides of eq w.r.t x.
- Step (3) Solve the resulting eq for \frac{dy}{dx}
Example 1: Explicit Differentiation
y+3x-5=0
Differentiate w.r.t x
\frac d{dx}\left(y+3x-5\right)=\frac d{dx}0
\frac d{dx}y+\frac d{dx}3x-\frac d{dx}5=0
\frac d{dx}y+3\frac d{dx}x-0=0
\frac d{dx}y+3=0
\boxed{\frac d{dx}y=-3}
Example 2:
Explicit Differentiation.
y^5+2x^3-3=0
Differentiate w.r.t x
\frac d{dx}\left(y^5+2x^3-3\right)=\frac d{dx}0
\frac d{dx}y^5+\frac d{dx}2x^3-\frac d{dx}3=0
5y^4\frac{dy}{dx}+2\frac d{dx}x^3-0=0
5y^4\frac{\displaystyle dy}{\displaystyle dx}+2(3x^2)=0
5y^4\frac{\displaystyle dy}{\displaystyle dx}+6x^2=0
5y^4\frac{\displaystyle dy}{\displaystyle dx}=-6x^2
\boxed{\frac{\displaystyle dy}{\displaystyle dx}=\frac{-6x^2}{5y^4}}
Example 3: 
Explicit Differentiation
y^2=\left(x+4\right)^\frac12
Differentiate w.r.t x
\frac d{dx}y^2=\frac d{dx}\left(x+4\right)^\frac12
2y\frac d{dx}y=\frac12\left(x+4\right).^{\frac12-1}\frac d{dx}\left(x+4\right)
2y\frac d{dx}y=\frac12\left(x+4\right).^{-\frac12}\left[\frac d{dx}x+\frac d{dx}4\right]
2y\frac d{dx}y=\frac12\left(x+4\right).^{-\frac12}\left[1+0\right]
2y\frac d{dx}y=\frac12\left(x+4\right).^{-\frac12}
\frac d{dx}y=\frac{\frac12\left(x+4\right).^{-\frac12}}{2y}
\frac d{dx}y=\frac{\left(x+4\right).^{-\frac12}}{4y}
\boxed{\frac d{dx}y=\frac1{4y.\left(x+4\right).^{-\frac12}}}
