Question

Find the derivative of the function:

Answer 1

you need a function for that

Answer 2

f(x)= [(4x^9)+(e^(3x+8))] / (5x^9) +2

Answer 3

explain steps

Answer 4

omg, why?

Answer 5

Thats gonna be a mess! :/

Answer 6

Possible derivation:
d/dx((4 x^9+e^(3 x+8))/(5 x^9)+2)
| Differentiate the sum term by term and factor out constants:
= | 1/5 (d/dx((4 x^9+e^(3 x+8))/x^9))+d/dx(2)
| The derivative of 2 is zero:
= | 1/5 (d/dx((4 x^9+e^(3 x+8))/x^9))+0
| Use the product rule, d/dx(u v) = v ( du)/( dx)+u ( dv)/( dx), where u = 1/x^9 and v = 4 x^9+e^(3 x+8):
= | 1/5 ((d/dx(4 x^9+e^(3 x+8)))/x^9+(4 x^9+e^(3 x+8)) (d/dx(1/x^9)))
| The derivative of 1/x^9 is -9/x^10:
= | 1/5 ((d/dx(4 x^9+e^(3 x+8)))/x^9+(4 x^9+e^(3 x+8)) (-9/x^10))
| Differentiate the sum term by term and factor out constants:
= | 1/5 ((4 (d/dx(x^9))+d/dx(e^(3 x+8)))/x^9-(9 (4 x^9+e^(3 x+8)))/x^10)
| Use the chain rule, d/dx(e^(3 x+8)) = ( de^u)/( du) ( du)/( dx), where u = 3 x+8 and ( de^u)/( du) = e^u:
= | 1/5 ((4 (d/dx(x^9))+e^(3 x+8) (d/dx(3 x+8)))/x^9-(9 (4 x^9+e^(3 x+8)))/x^10)
| The derivative of x^9 is 9 x^8:
= | 1/5 ((e^(3 x+8) (d/dx(3 x+8))+4 (9 x^8))/x^9-(9 (4 x^9+e^(3 x+8)))/x^10)
| Differentiate the sum term by term and factor out constants:
= | 1/5 ((e^(3 x+8) (3 (d/dx(x))+d/dx(8))+36 x^8)/x^9-(9 (4 x^9+e^(3 x+8)))/x^10)
| The derivative of 8 is zero:
= | 1/5 ((e^(3 x+8) (3 (d/dx(x))+0)+36 x^8)/x^9-(9 (4 x^9+e^(3 x+8)))/x^10)
| The derivative of x is 1:
= | 1/5 ((36 x^8+3 1 e^(3 x+8))/x^9-(9 (4 x^9+e^(3 x+8)))/x^10)

Answer 7

how did you type that so fast?

Answer 8

Lol he copied it from Wolframalpha

Answer 9

but it does explain it to you
http://www.wolframalpha.com/input/?i=%5B%284x%5E9%29%2B%28e%5E%283x%2B8%29%29%5D%20%2F%20%28%285x%5E9%29%20%2B2%29&t=mfftb01

Answer 10

Scroll down to "derivative" and there is a "show steps" button.

Answer 11

\[y=\frac{4x^9+e^{3x+8}}{5x^9+2}\]
Take natural log of both sides!
\[\ln(y)=\ln(\frac{4x^9+e^{3x+8}}{5x^9+2})\]
Using properties of log we can rewrite the left hand side as follows:
\[\ln(y)=\ln(4x^9+e^{3x+8})-\ln(5x^9+2)\]
Now take derivative of both sides to find y':
\[\frac{y'}{y}=\frac{4 \cdot 9 \cdot x^{9-1}+(3x+8)'e^{3x+8}}{4x^9+e^{3x+9}}-\frac{5 \cdot 9 \cdot x^{9-1}+0}{5x^9+2}\]
so we have
\[\frac{y'}{y}=\frac{36x^{8}+(3+0)'e^{3x+8}}{4x^9+e^{3x+9}}-\frac{45x^{8}}{5x^9+2}\]
\[\frac{y'}{y}=\frac{36 x^{8}+3e^{3x+8}}{4x^9+e^{3x+9}}-\frac{45 x^{8}}{5x^9+2}\]
To find y' multiply y on both sides:
\[y'=y (\frac{36 x^{8}+3e^{3x+8}}{4x^9+e^{3x+9}}-\frac{45 x^{8}}{5x^9+2})\]
and we replace y with \[\frac{4x^9+e^{3x+8}}{5x^9+2}\]