Differentiation Chain Rule

The chain rule is a calculus technique to differentiate a function, which may consist of another function inside it. Let us say the function g(x) is inside function f(u), then you can use substitution to separate them in this way. Each function then differentiates separately, and you need to find the derivatives du/dx and dy/du.

Once you have the derivatives, you simply multiply them to give you the result. The du terms obviously cancel out.

Chain Rule Example

Here is a simple binomial function y=(x+1)² which you can easily differentiate by multiplying out the bracket. However, in this example the chain rule works as well.

The first step is to use substitution to separate out the two functions. If u = x+1 then y = u² which is simple to understand.

If you differentiate (x+1) the result is 1. Therefore du/dx=1

If you differentiate y=u² then dy/du = 2u. We also know that u =x+1 so we can substitute that back in to give dy/du = 2x+2

Since dy/dx = (du/dx) × (dy/du), this simply means that you have to multiply the derivatives and the du terms cancel out, and as you can see, the result is 2x+2.

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