What is Chain Rule
Chain rule is one of the core principles in calculus we're going to use it anytime you take the derivative of anything even reasonably complex and it's called the chain rule derivative.
When you're first exposed to it, it can seem a little daunting and a little bit convoluted but as you see more and more chain rule examples it'll start to make sense and hopefully you'll even start to seem a little bit simple.
Let's move on to the chain rule we're going to cover a lot of examples foe understanding the chain rule differentiation. The first form where you need to be familiar with is the derivative of composite function.
$$ \frac{d}{dx} (f[g(x)])= f'[g(x)] \cdot g'(x) $$
A composite function is run where you have one function inside of another and notice that g(x) is inside of f(x) which makes it a composite function.
That's the main idea behind the chain rule derivatives. If you follow this process you can get the answer easily.
Process of Chain Rule Differentiation
Let’s talk in general chain rule example
Say we have a function
$$ h(x) = (sin x)^2 $$
Now, what will be the derivative of composite function h(x) i.e. h'(x) = ?
Means we have to work for dh/dx, so for this we will use the chain rule derivatives.
The chain rule differentiation comes into play every time any time your function can be used as a composition of more than one function.
I ask derivative of x2 w.r.t "x"
$$ \frac{d}{dx}(x^2)=2x $$
Similarly, I ask derivative of a2 w.r.t "a"
$$ \frac{d}{da}(a^2)=2a $$
Now, you can analyse what will be the derivative of (sin x)2 w.r.t "sin x"
$$ \frac{d}{d(sin x)}(sin x)^2= 2(sin x) $$
So the chain rule tells us that this derivative of the whole function w.r.t inner function times the derivative of inner function w.r.t x.
same as we stated chain rule above,
$$ \frac{dh}{dx}=2sinx \cdot \frac{d}{dx}(sin x) $$ $$ \frac{dh}{dx} = 2sinx \cdot cosx $$
This shows, derivative of the whole function w.r.t inner function multiplied with the derivative of inner function w.r.t x, and that is called the chain rule for finding derivatives.
I.e. The Chain Rule formula is represented as:
$$ \frac{d}{dx} (f[g(x)])= f'[g(x)] \cdot g'(x) $$
In general, we can write chain rule as below:
$$ \frac{d}{dx} [u^n] = n(u)^{n-1} \frac{d}{dx}(u) $$
Chain Rule Examples
So, Let's work on some useful examples of chain rule which will polish our concept of the derivative of composite functions. Let's try to implement chain rule formula in following examples:
Example:
Consider a function given as :
$$ y = (5x+3)^4 $$
Then derivative according to the chain rule is:
$$ \frac{d}{dx}(5x+3)^4 = 4(5x+3)^{4-1} \cdot \frac{d}{dx}(5x+3) $$ $$ \frac{d}{dx}(5x+3)^4 = 4(5x+3)^3 \cdot 5 $$ $$ \frac{d}{dx}(5x+3)^4 = 20(5x+3)^3 $$
Example:
Consider a function given as :
$$ y = (ln x)^7 $$
The derivative of composite function is:
$$ \frac{d}{dx}(ln x)^7 = 7(ln x)^{7-1} \frac{d}{dx}(ln x) $$ $$ \frac{d}{dx}(ln x)^7 = 7(ln x)^6 \cdot \frac{1}{x} $$ $$ \frac{d}{dx}(ln x)^7 = \frac{7(ln x)^6}{x} $$
Example:
Consider a function given as :
$$ y = sin[tan(x^4)] $$
Solution:
Now here there is a composite function inside a composite function, there is not a rocket science we have to solve it using the same steps:
I.e derivative of outer function multiplied by inner function’s derivative and also multiplied by the derivative of most inner function w.r.t x.
$$ \frac{d}{dx}sin[tan(x^4)]= cos[tan(x^4)] \cdot \frac{d}{dx}tan(x^4) \cdot \frac{d}{dx}x^4 $$ $$ \frac{d}{dx}sin[tan(x^4)]= cos[tan(x^4)] \cdot sec^2(x^4) \cdot 4x^3 $$