How do I find derivatives involving natural logarithms and multiple variables?

[Dustobusto]

So I have an exam tomorrow, and the teacher provided a review.

f(x) = ln(x + y)

I remember that

d/dx ln[f(x)] = f'(x)/f(x) so would that not equal 2/(x + y) ? The answer she gave is

1/(x + y - 1) ... where that neg. one came from I have no idea. Come to think of it, there were no problems on the homework that included two variables in this manner, so maybe the properties are slightly different?

Also, find the derivative of sin(x)^{ln 3x}... another problem that doesn't bear much resemblance to anything on the homework, a little confused on the order of operations here.

I know that with something like 5^3x would be (3 ln 5)5^3x,

so for this one, b = sin (x). Trying to replicate it the way the book would,

(ln sin x)sin(x)^{ln 3x} * (ln 3x)'

the derivative of ln(3x) = 3/3x, bring that to the front and get ...

I guess what I'm asking is do I use the general power rule or was I on the right track? At any point in time am I supposed to take the derivative of sin(x) and turn it into cos x?

 

[CAF123]

So I have an exam tomorrow, and the teacher provided a review.

f(x) = ln(x + y)

I remember that

d/dx ln[f(x)] = f'(x)/f(x) so would that not equal 2/(x + y) ? The answer she gave is

1/(x + y - 1) ... where that neg. one came from I have no idea. Come to think of it, there were no problems on the homework that included two variables in this manner, so maybe the properties are slightly different?

Both are incorrect and given that you have a function of x only, ie f(x) you take y as a constant (so it is still a one variable problem).

 

[Fredrik]

I don't know where she got that -1, but I also don't know where you got the 2. y denotes a real number here, right? Not another function?

If there seems to be no way to use any of the standard rules for derivatives (the rules for sums, products, quotients, inverses and compositions), the trick is often to use the rules for exponentials to rewrite the expression you have so that you can use those rules. For example,
$$\frac{d}{dx}x^x=\frac{d}{dx}e^{\log x^x}=\cdots$$ A similar trick should work with the problem you're asking about. Another approach is to rewrite sin in terms of exponentials (and complex numbers).

I don't understand what you're saying about $5^{3x}$.

 

[scurty]

You can also use implicit differentiation: write the equation as $y = sin(x)^{ln(3x)}$ and take the log of both sides. Solve for $dy/dx$.