Explicit Differentiation

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:

ExplicitDifferentiation

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}}}