Chain Rule
What is the Chain Rule?
The chain rule is used to find the derivative of composite functions. Let’s say you have a function within another function. The chain rule will allow us to find the derivative of this.
First, a composite function takes the form:
[latex]F(x) = f(g(x))[/latex] or [latex]F = f \circ g [/latex]
This essentially means that F(x) is a composite function, where f(x) is the outer function and g(x) is the inner function. This rule will help us find the derivative of F(x). See the following definition:
Chain Rule:
If [latex]F(x) = f(g(x))[/latex] where [latex]f(x)[/latex] and [latex]g(x)[/latex] are both differentiable functions, then the derivative of F(x) is:
[latex]F'(x) = f'(g(x))g'(x)[/latex]
Basically, you first multiply the derivative of the outside function with the inside function left alone, and with the derivative of the inner function.
Let us now use the chain rule to find the derivative of the following function:
[latex]F(x) = (2x^3+5)^2[/latex]
First, we identify that the inner function is [latex]f(x) = 2x^3+5[/latex] and the outer function is the power of 2. As per the power rule as discussed before, to find the derivative of the outer function,, we will bring the 2 down and then subtract the exponent by 1.
Then, again using the power rule, the derivative of the inner function is [latex]6x^2[/latex]. Now, multiplying it all together using the chain rule, we have:
[latex]F'(x) = 2(2x^3+5)(6x^2)[/latex]
[latex]F'(x) = 24x^5 + 60x^2[/latex]
As shown, the chain rule is a powerful tool which allows us to find the derivative of composite functions.
Example
Find the derivative of the following function:
[latex]\Large{F(x) = sin(x^2+3)}[/latex]
First, we need to identify that the outer function is the sine function, and the inner function is [latex]x^2+3[/latex].
Then, based on the chain rule as we discussed, we can find the derivative as follows:
[latex]\large{F'(x) = cos(x^2+3) \cdot \frac{d}{dx}(x^2+3)}[/latex]
The derivative of sin(x) is cos(x), but we need to substitute the x for the inner function, and we must also multiply by the derivative of the inner function, as we just shown above.
So, we have:
[latex]\large{F'(x) = 2xcos(x^2+3)}[/latex]
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