What is the derivative of $\frac{4 x^{8} - \sqrt{x}}{8 x^{4}}$ $(4x^8-sqrt(x))/(8x^4)$ ?

Question

2681 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2015-08-01T00:34:40

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

Start by rewriting your function like this

$y = (4x^color(red)(cancel(color(black)(8))))/(8color(red)(cancel(color(black)(x^4)))) - x^(1/2)/(8x^4)$

$y = 1/2x^4 - 1/8 * x^(-7/2)$

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 can be rewritten as

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

What is the derivative of (4x^8-sqrt(x))/(8x^4)4x8−√x8x4(4x^8-sqrt(x))/(8x^4)?