How do you use the chain rule to differentiate #y=(-3x^5+1)^3#?

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Answer 1 · 2018-04-14T12:45:08

#(dy)/(dx)= -45x^4(-3x^5+1)^2#

#y=(-3x^5+1)^3#

#(dy)/(dx)= 3(-3x^5+1)^2times(-15x^4)#

#(dy)/(dx)= -45x^4(-3x^5+1)^2#

Answer 2 · 2018-04-14T12:47:49

#d/dx[(-3x^5+1)^3]-45x^4(-3x^5+1)^2#

The chain rule states that:

#d/dx[f(g(x))]=f'(g(x))*g'(x)#

Hmm... What does that mean?

To use the chain rule, we need to find the inside function and the outside function.

The inside function is #(-3x^5+1)#

The outside function is #x^3#

Using the power rule:

#d/dx[x^n]=nx^(n-1)#

We find the derivative of #x^3#

#=>3*x^(3-1)#

#=>3x^2#

We do the similar thing with the inside function.

#=>-3*5x^(5-1)+1*0*x^(0-1)#

#=>-15x^(4)+0#

#=>-15x^(4)#

Now, we put the original inside function inside the derivative of the outside function.

#=>3*(-3x^5+1)^2#

We multiply this by the derivative of the inside function.

#=>3*(-3x^5+1)^2*-15x^(4)#

#=>-45x^4(-3x^5+1)^2# That is the answer!