Explicit Differentiation and its Examples with Solution

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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:

$y+3x-5=0$

$y^5+2x^3-3=0$

$y^2=left(x+4right)^frac12$

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 $x$

$frac d{dx}left(y+3x-5right)=frac d{dx}0$

[sum and difference rule]

$frac d{dx}y+frac d{dx}3x-frac d{dx}5=0$

$frac d{dx}y+3frac d{dx}x-0=0$

$frac d{dx}y+3=0$

$boxed{frac d{dx}y=-3}$

Example 2:

Explicit Differentiation.

$y^5+2x^3-3=0$

Differentiate w.r.t $x$

$frac d{dx}left(y^5+2x^3-3right)=frac d{dx}0$

[sum and difference rule]

$frac d{dx}y^5+frac d{dx}2x^3-frac d{dx}3=0$

$5y^4frac{dy}{dx}+2frac d{dx}x^3-0=0$

$5y^4frac{displaystyle dy}{displaystyle dx}+2(3x^2)=0$

$5y^4frac{displaystyle dy}{displaystyle dx}+6x^2=0$

$5y^4frac{displaystyle dy}{displaystyle dx}=-6x^2$

$boxed{frac{displaystyle dy}{displaystyle dx}=frac{-6x^2}{5y^4}}$

Example 3:

Explicit Differentiation

$y^2=left(x+4right)^frac12$

Differentiate w.r.t $x$

$frac d{dx}y^2=frac d{dx}left(x+4right)^frac12$

$2yfrac d{dx}y=frac12left(x+4right).^{frac12-1}frac d{dx}left(x+4right)$

$2yfrac d{dx}y=frac12left(x+4right).^{-frac12}left[frac d{dx}x+frac d{dx}4right]$

$2yfrac d{dx}y=frac12left(x+4right).^{-frac12}left[1+0right]$

$2yfrac d{dx}y=frac12left(x+4right).^{-frac12}$

$frac d{dx}y=frac{frac12left(x+4right).^{-frac12}}{2y}$

$frac d{dx}y=frac{left(x+4right).^{-frac12}}{4y}$

$boxed{frac d{dx}y=frac1{4y.left(x+4right).^{-frac12}}}$

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