Chain Rule

Chain Rule: Chain Rule is differential rule to evaluate the inner functions of the given functions.

Chain Rule Examples

Example y= sin(cosx) , y= log ( tanx). Here cosx is the inner function of sin and tanx is the inner function of log

Chain Rule Example

2. Find the derivative of x for y = log(sinx)=$\frac{\mathrm{d} log(sinx)}{\mathrm{d}}$ x $\frac{1}{sinx}$cosx = tanx.

Explanation: First write the derivative of log with inner term and then differentiate the inner function sinx.

3. Find the derivative of y = tan ( secx + cosx)

$\frac{\mathrm{d} tan(secx+cosx)}{\mathrm{d} x}$= $sec^{2}$(secx+cosx)$\times (secxtanx-sinx)

Explanation: First write the derivative of tan with inner term and then differentiate the inner function secx + cosx

For the above example given we can substitute the inner one as u . That is y= sinu so $\frac{\mathrm{d} y}{\mathrm{d} x}$= $\frac{\mathrm{d} y}{\mathrm{d} u}$$\times$$\frac{\mathrm{d} u}{\mathrm{d} x}$.


Chain Rule of Differentiation

1. Find the derivative y=$cos(3x^{4}+8)^2$


Step1: Let $(3x^{4}+8)^2$ =u

So, y= $cosu^2$

we have the formulae $\frac{\mathrm{d} y}{\mathrm{d} x}$=$\frac{\mathrm{d} y}{\mathrm{d} u}$$\times$$\frac{\mathrm{d} u}{\mathrm{d} x}$.

$\frac{\mathrm{d} y}{\mathrm{d} x}$= $\frac{\mathrm{d} cosu}{\mathrm{d} u}$$\times$$\frac{\mathrm{d} (3x^{4}+8)^2 }{\mathrm{d} x}$

step2: sinu$\times$ $2(3x^4)$$\times$ $12x^3$.

Explanation: The derivative of cosu is sinu and the derivative of $u^2$ is 2u and the inner most term is

$x^4$ so the derivative is $4 x^3$.

Next Chapters

Number Theory Linear Equation Set Theory Math Fractions
Math Functions Pyramid Calculus Cone
Cylinder Chain Rule Limits and Continuity Prime Factorization
Square Roots and Cube Roots Parabola Distance Formula Definite Integrals
Interest Simple Interest Compound Interest Area of Irregular Figures

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