Differentiate the problem using Logarithmic Differentiation $$y=(x+6)^x$$

Question

juscallmesteve · Answer

When I do this I get 

$$y \prime=(x+6)^x(\frac{ x }{ (x+6) }+\ln(x+6))$$

juscallmesteve · Answer

also how do i give people medals for helping me?

solomonzelman · Answer

To give user a medal, click Best Response next to their reply.
Now, as far as the differentiation process here goes...

juscallmesteve · Answer

O i did not know that game someone a medal

solomonzelman · Answer

In general,

$\color{#000000}{    \displaystyle   y=f(x)^{g(x)}        }$

$\color{#000000}{    \displaystyle   \ln y=\ln \left(f(x)^{g(x)}\right)        }$

$\color{#000000}{    \displaystyle   \ln y=g(x)\ln \left(f(x)\right)        }$

differentiate both sides

$\color{#000000}{    \displaystyle  \frac{y'}{y}=g'(x)\ln \left(f(x)\right)+g(x)\frac{f'(x)}{f(x)}        }$

$\color{#000000}{    \displaystyle y'=y\cdot \left(g'(x)\ln \left(f(x)\right)+g(x)\frac{f'(x)}{f(x)}  \right)      }$

$\color{#000000}{    \displaystyle y'=f(x)^{g(x)} \cdot \left(g'(x)\ln \left(f(x)\right)+g(x)\frac{f'(x)}{f(x)}  \right)      }$

this is the process in general.

solomonzelman · Answer

and your result is correct.

solomonzelman · Answer

(so I guess there wasn't a need going over that process once again)

juscallmesteve · Answer

Thank you very much for all of that information

solomonzelman · Answer

Not a problem!

solomonzelman · Answer

good job!