__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$\times$ $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 $4 x^3$.

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