Post #4 The Chain Rule

CHAIN RULE!

What is the chain rule?

            The chain rule is used to find the derivative of a composition of two or more functions.

If f and g are functions the chain rule shows this

            (f*g)’= (f’*g) * g’ in Leibniz notation it is expressed as dz/dx=dz/dy*dy/dx

To better understand what the chain rule does let’s try an example

            Let f(x)= 6x+3 and g(x)= -2x+5

            Solve for f(x) and g(x) separately using the other derivative rules

            f(x)= 6

            g(x)= -2

            let’s say the derivative of the two functions is h’(x)

            using the chain rule: h’(x)=f’(g(x))g’(x)

            so… =f’(-2x+5)(-2)

            to simplify 6(-2)= -12

            we got -12 by plugging g(x) into the derivative of f;f’

This is an easier example of the chain rule because f’=is a constant value.

Now let’s try another example.

            Let f(x) =x^7 and g(x) = (x^2+3)

            so we again want to solve each separately

            use product rule to solve

            f(x)= 7x^6

            g(x)= 2x

            now we will name the derivative H’(x)

            f(g(x))*g’(x)

            input g(x) into f(g)) as X

            f(g(x))= (x^2+3)^7

            Now we want to include f’ and g’ in our function H’(x)

            7(x^2+3)^6 *2x

            g(x) was added into the formula where x in f(x) was.

            Next we need to simplify the formula

            H’(x)= 14x(x^2+3)^6

            The chain rule is important because it lets you solve equations when they contain more than one function!

For further study here are some example problems to try out and there solutions are listed below!

            f(x)=sin(x) and g(x)= 2x

             f(x)=√X  and g(x)=5z-8

            f(x)= sinx and g(x)=(3x^2+x)

Answer A. H’(x) =2cos (2x)

Answer B. H’(x) = 5/(2√(5z-8))

Answer C. H’(x) = cos( 3x^2+x)(6x+1)