# 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

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$Explanation: Step1: Let$(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}$step2: sinu$\times2(3x^4)\times12x^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\$.

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