Need help with derivative of logs. Problem in comments

Question

anonymous · Answer

$$f(x)=\sin(3*\ln(x))$$

solomonzelman · Answer

derive using the chain rule.

solomonzelman · Answer

derive first the sin(3lnx) as if it was just sin, so you get
cos(3lnx)      (this, the derivative of sin, can be proven, but lets accept that as given)

then multiply times the derivative of the inner function, times the derivative of ln3.

What is the derivative of ln3 ?

solomonzelman · Answer

I mean times the derivative of  3lnx, sorry.

anonymous · Answer

1/3?

solomonzelman · Answer

the derivative of ln(x) is ?

solomonzelman · Answer

Need help doing the derivative of ln(x) ?

anonymous · Answer

yes

solomonzelman · Answer

$\large\color{black}{  y=\ln(x)    }$

PROVE.

$\large\color{black}{  y=\log_e(x)    }$

$\large\color{black}{  e^y=x    }$   (finding dy/dx)

$\large\color{black}{ \frac{dy}{dx} e^y=1    }$

$\large\color{black}{ \frac{dy}{dx} =1/e^y    }$

we know e^y=x, so,     $\large\color{black}{ \frac{dy}{dx} =1/x    }$

solomonzelman · Answer

the derivative of ln(x) is 1/x.

solomonzelman · Answer

And therefore, the derivative of 3ln(x) is?

anonymous · Answer

3/x?

solomonzelman · Answer

yes, exactly:)

solomonzelman · Answer

So the derivative of sin(3ln(x))  (of function y, with respect to x) is as follows

$\large\color{black}{    cos(3 \ln x) \times \frac{3}{x}  }$

solomonzelman · Answer

we differentiated it using the chain rule, the first part is the derivative of the sin(3lnx) as if ti is just sin(x), but using chain rule, multiplied times the derivative of the inner function, of 3ln(x), which we know to be 3/x.

anonymous · Answer

Okay it makes sense now. So is that all I have to do?

solomonzelman · Answer

yes, so your answer is

$\LARGE\color{black}{     \frac{3cos(3 \ln x)}{x}  }$

solomonzelman · Answer

makes sense?

anonymous · Answer

Yes the chain rule is sometimes confusing but it's making more sense now. Thank you :)

solomonzelman · Answer

You welcome:)