Question

Please help me to determine
d/dx(4√x-8/√x+πx^3)

Answer 1

The best way to do such problems is to do a substitution (at least when you're starting)
u = 4√x-8
v = √x+πx^3
What is (u/v)' ?

Answer 2

Hint: (1/v)' = -1/v^2
Now use the product rule
Or look up the quotient rule.

Answer 3

telliott99 can you go further please?

Answer 4

Find \[\frac{d}{dx} (\frac{4 \sqrt{x} - 8}{\sqrt{x} + \pi x^3})\]
Calculate the derivatives of numerator and denominator
\[\frac{d}{dx} (4 \sqrt{x} - 8) = \frac{2}{\sqrt x}\]
\[\frac{d}{dx} ({\sqrt{x} + \pi x^3}) = \frac{1}{2 \sqrt x} + 3 \pi x^2\]
Use the quotient rule
\[(u/v)' = (v u' - u v')/v^2\]
Substituting, the numerator is:
\[(\sqrt{x} + \pi x^3)(\frac{2}{\sqrt x}) - (4 \sqrt {x} - 8) (\frac {1}{2 \sqrt(x)} + 3 \pi x^2)\]
And the denominator is:
\[(\sqrt {x} + \pi x^3)^2\]
After that it's just algebra. Ugly, but algebra.