Whether it is Calculus AB or Calculus BC, being completely fluent in taking the derivative of almost any function is imperative.
At minimum, you should know how to use:
• The chain rule
• The power rule
• The product rule
• The quotient rule

Chain rule comes into picture when we have composite function. Question is, What is a composite function ?

Composite Function :

Call it a composite function, or a function of functions, or nested functions etcetera, I prefer to call it “composition” of functions – as in, a mixture of various functions mingled among each-other.
Usually, it is represented as:

    \[ f(g(x)) \]

For example:


In short, we have a function inside a function.
In this case, x^2 is inner function while tangent is outer function

Chain rule :

We have now an understanding of composite functions. Let’s come to business, how to differentiate it ?
Here’s how it is:

    \[ f(g(x))'=f'(g(x)g'(x) \]

Let’s understand what’s going on here.

We differentiated the first guy f(x) keeping g(x) as it is, then we went inside and differentiated g(x), then we multiplied both these.

Example :

    \[\frac{dsinx^2}{dx}  =  cosx^2 \frac{dx^2}{dx} = 2xcosx^2  \]

Power-chain rule :

Whenever we have a power raised to a composite function , we differentiate using the power rule first, and then apply chain rule

    \[\frac{d(x^2+2x)^2}{dx} = 2(x^2+2x)\frac{d(x^2+2x)}{dx} = 2(x^2+2x)(2x+2) \]

Implementing the chain rule is usually not difficult.  The problem that many students have trouble with is trying to figure out which parts of the function are within other functions. That’s where I recommend an informal approach to chain rule:

Informal approach:

Even though few people admit it, almost everyone thinks along the lines of the informal approach as shown:


    \[f(x) = (stuff)^n \]


    \[\frac{df}{dx} = n(stuff)^{(n-1)}  \frac{d(stuff)}{dx}\]

You will usually see this written as :

    \[\frac{du^n}{dx}=nu^{(n-1)} \frac{du}{dx}\]

Closing words :

I personally regard differentiation as destruction (and integration as construction ) .

Treat every composite function as a building which we want to bring down. While breaking, you will obviously start from the outermost part and will start breaking it part by part and will move inside.

That is chain rule.

Example: sin(cos(lnx))

Break sin first keeping following stuff same

Break cos keeping following stuff same

And then break lnx

Multiply all

And voila!

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