Thread: Logarithmetic Differentiation?

1. Logarithmetic Differentiation?

Find the derivative of the function.

I followed an example in the book (the only one that covers this) and ended up with this

what am i doing wrong?

2. Originally Posted by masterofcheese

Find the derivative of the function.

I followed an example in the book (the only one that covers this) and ended up with this

what am i doing wrong?

$\displaystyle ln(y) = ln(\sqrt{x}e^{x^2}*(x^2+2)^{11})$
$\displaystyle ln(y) = ln(\sqrt{x}e^{x^2}) + 11ln(x^2+2)$

Derive. Left side goes to $\displaystyle \frac{y'}{y}$ and the right side involves the Chain Rule.

Once you've derived the entire right side multiple by y (your original function) and you'll have y'.

3. Originally Posted by masterofcheese

Find the derivative of the function.

I followed an example in the book (the only one that covers this) and ended up with this

what am i doing wrong?

Basically, all. Now, you have here the product of three functions and derivating that using the product formula is very boring, so you better use logarithms to turn multiplication into sum and derivate implicitly (noting, btw, that the function is always non-negative and in fact positive for non-zero x):

$\displaystyle \ln y= \ln \sqrt{x}+\ln e^{x^2}+\ln (x^2+2)^{11}=\frac{1}{2}\ln x + x^2 + 11\ln(x^2+2)\Longrightarrow$ $\displaystyle \frac{1}{y}dy=\left(\frac{1}{2x}+2x+\frac{22x}{x^2 +2}\right)dx\Longrightarrow \frac{dy}{dx}=y\left(\frac{1}{2x}+2x+\frac{22x}{x^ 2+2}\right)$

Tonio