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Chain Rule

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Chain Rule is an important study with in calculus. In differentiation, the chain rule is the formula for finding the derivative of the composite of two or more functions.

In integration, chain rule is defined as the U-Substitution which is the counter part of the chain rule. The formula for processing the derivative of the composition of some functions.

The chain rule have first been used by Leibniz. The chain rule is usually applied to composites in excess of two functions. Quotient rule is a consequence of the chain rule along with the product rule.
Consider f and g for being functions then, (f o g)' = (f' o g). g'.
In integration, the counterpart to the chain rule is the substitution rule. Also exists for differentiating a function of another function.
To differentiate y = $f(g(x))$, let $u$ = $g(x)$, then $y$ = $f(u)$ and

$\frac{dy}{dx}$  = $\frac{dy}{du}$ * $\frac{du}{dx}$

Chain Rule Formula

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The chain rule is usually an important method in calculus to unravel the derivatives. This method is needed to compute the derivatives having several functions. Say, if you can find two functions , f and g, then in line with chain rule the derivative from the term f in addition to g is expressed since the derivative of the composite function fog.
The Chain Rule Formula is stated since:
$\frac{dy}{dx}$ = $\frac{dy}{du}$. $\frac{du}{dx}$

Here, the differention of 'y' with respect to 'x' is equal to the differentiaton of 'u' with respect to 'x' times of differentiation of 'y' with respect to 'u'.

Examples

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Given below are some of the examples on chain rule:
Example 1: Find your derivative of y = $(5x+5)^2 $.

Remedy: We can utilize the chain rule to unravel this problem. Your derivative of y = derivative of $(5x+5)^2 $. We could write this while, derivative of y = 2 (5x + 5) $\times$ $\frac{d}{dx}$ (5x+5). This equals to the derivative of y = 2 (5x + 5) $\times$ 5 = 10 (5x + 5).

Example 2: Find your derivative of y =tan ($e^x $).

Remedy: We know, your derivative of y = derivative of tan ($e^x $). We could write this while, derivative of y = sec$^{2}$ ($e^x $) derivative of ($e^x $). This equals to the derivative of y = $e^ x$ sec$^{2}$ ($e^x $).

Example 3: Find your derivative of y = sin (9x).

Remedy: We know, your derivative of y = derivative involving sin (9x) compatible to cos (9x) times derivative of (9x). This add up to 9 times cos (9x).

Product Chain Rule

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With all the product rule in addition to chain rule to obtain the derivative of some sort of composite function is referred to as the product cycle rule. In this particular rule, we can operate the product rule along with the chain rule together.

Given below is an example:
Example 1: Find the derivative of $x^{3}(2x^{2} + 3)$

Solution: Let $y = f(x) = x^{3}(2x^{2} + 3)$

Then, $y' = f'(x) = x^{3}2(2x) + (2x^{2} + 3)(3x^{2})$

= $4x^{4} + (3x^{2}) (2x^{2} + 3)$

= $4x^{4} + 6x^{4} + 9x^{2}$

= $10x^{4} + 9x^{2}$

= $x^{2}(10x^{2} + 9)$

Example 2: Differentiate y = (3x + 1)$^{2}$

Solution: For the given problem first differentiate the square of (3x + 1) unchanged. Again then consider differentiating (3x + 1)

Thus D(3x + 1)$^{2}$ = 2(3x + 1) D(3x + 1)

= 2(3x + 1) . 3

= 6(3x + 1)

Example 3: Using chain rule, solve the below problem.

f(x) = 6x + 3 and g(x) = -2x + 5, find h'(x).

Use $h(x) = f(g(x))$

Solution: The derivatives of f and g are

$f'(x)$ = 6 and $g'(x)$ = - 2

Now according to the chain rule,

$h'(x) = f'(g(x))g'(x)$

= f'(-2x + 5) (- 2)

= 6(-2)

= -12

$h'(x) = f(g(x))$

=f( -2x + 5)

= 6( -2x + 5) + 3

= -12x + 30 +3

= -12x + 13

Therefore, the solution is $h'(x)$ = - 12.