SOLUTION: Determine the derivative of the following. 1.y= loga(x^3+2x-2) 2.y= ln^3(x+ sin x)

Algebra ->  Test -> SOLUTION: Determine the derivative of the following. 1.y= loga(x^3+2x-2) 2.y= ln^3(x+ sin x)       Log On


   

Question 1193579: Determine the derivative of the following.
1.y= loga(x^3+2x-2)
2.y= ln^3(x+ sin x)
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
1.
y=+log%28a%2C%28x%5E3%2B2x-2%29%29

First change the log base a to "ln" (which is log base e) by 
replacing "loga" by "ln" and dividing by ln(a).

y=+ln%28x%5E3%2B2x-2%29%2Fln%28a%5E%22%22%29

ln%28a%5E%22%22%29%2Ay=ln%28x%5E3%2B2x-2%29

Use the ln derivative formula: "(The derivative of what the "ln" is of)
divided by what the "ln" is of.

ln%28a%5E%22%22%29expr%28dy%2Fdx%29=%283x%5E2%2B2%29%2F%28x%5E3%2B2x-2%29

dy%2Fdx=%283x%5E2%2B2%29%2F%28ln%28a%29%28x%5E3%2B2x-2%29%29

-------------------------------------------

2.
y=+ln%5E3%28x%5E%22%22%2B+sin%28x%29%29

Write it so you can tell what's being raised to the 3rd power:

y=+%28ln%28x%5E%22%22%2B+sin%28x%29%29%5E%22%22%29%5E3

Use the power rule and the chain rule



Use the ln derivative formula: "(The derivative of what the "ln" is of)
divided by what the "ln" is of.



Simplify the appearance of the first factor by putting the 2
exponent immediately after the "ln"



Edwin