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

2 Answers
Apr 14, 2018

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

Explanation:

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

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

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

Apr 14, 2018

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

Explanation:

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!