What is the derivative of #(4x^8-sqrt(x))/(8x^4)#?

Answer 1

#y^' = 1/16 * (32x^(15/2) + 7) * x^(-9/2)#

To begin, rewrite your function as follows:

#y = (4x^color(red)(cancel(color(black)(8))))/(8color(red)(cancel(color(black)(x^4)))) - x^(1/2)/(8x^4)#
#y = 1/2x^4 - 1/8 * x^(-7/2)#
Now you can use the power rule to differentiate #y#
#y = 1/2[d/dx(x^4)] - 1/8 d/dx(x^(-7/2))#
#y = 1/color(red)(cancel(color(black)(2))) * color(red)(cancel(color(black)(4))) x^3 - 1/8 * (-7/2) * x^(-9/2)#
#y = 2x^3 + 7/16x^(-9/2)#

This could be rephrased as

#y^' = (x^(-9/2) * (16 * 2x^(15/2) + 7))/16 = color(green)(1/16 * (32x^(15/2) + 7) * x^(-9/2))#
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

To find the derivative of the function ((4x^8 - \sqrt{x})/(8x^4)), we can use the quotient rule. The quotient rule states that if (u(x)) and (v(x)) are differentiable functions, then the derivative of their quotient is ((u'v - uv')/(v^2)). Applying the quotient rule to the given function, we have:

[ f(x) = \frac{4x^8 - \sqrt{x}}{8x^4} ]

where (u(x) = 4x^8 - \sqrt{x}) and (v(x) = 8x^4). Now, differentiate (u(x)) and (v(x)) with respect to (x):

[ u'(x) = 32x^7 - \frac{1}{2\sqrt{x}} ]

[ v'(x) = 32x^3 ]

Now, apply the quotient rule:

[ f'(x) = \frac{(32x^7 - \frac{1}{2\sqrt{x}})(8x^4) - (4x^8 - \sqrt{x})(32x^3)}{(8x^4)^2} ]

Simplify this expression to get the derivative of the given function.

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