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

Answer 1

#(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#
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

#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!
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 3

To differentiate ( y=(-3x^5+1)^3 ) using the chain rule, follow these steps:

  1. Identify the inner function ( u ) as ( -3x^5+1 ).
  2. Find its derivative ( \frac{du}{dx} ), which is ( -15x^4 ).
  3. Raise the inner function to the power of 2, yielding ( u^2 = (-3x^5+1)^2 ).
  4. Multiply by the derivative of the inner function: ( 2(-3x^5+1)(-15x^4) ).
  5. This gives the derivative of the outer function with respect to the inner function: ( \frac{d}{du} [u^3] = 3u^2 ).
  6. Substitute ( u = -3x^5+1 ) back in: ( 3(-3x^5+1)^2 ).
  7. Finally, multiply by the derivative of the inner function: ( 3(-3x^5+1)^2 \cdot (-15x^4) ).

So, the derivative of ( y=(-3x^5+1)^3 ) is ( \frac{dy}{dx} = 3(-3x^5+1)^2 \cdot (-15x^4) ).

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7