I am assuming that you use '*' for multiplication.
You can notice that, taking derivative of summation is 'easier' than those of multiplication.
Say: (u + v + w)' = u' + v' + w' looks far easier compared with (uvw)' = u'vw + uv'w + uvw', right?
And using log function, one can change multiplication, into summation. Hence, making it more comfortable when dealing with a bunch of multiplication.
danielle36 said:
Hello,This is my first crack at using log differentiation, but I can't seem to get too far with it...
Use logarithmic differentiation to calculate the derivative for the following function:
y = \sqrt{x}e^{x^{2}} (x^{2} + 1)^{10}
lny = \frac{1}{2}lnx * x^{2}lne * 10ln(x^{2} + 1)
Nah, you are wrong from the first line :(, you should note that:
ln(xy) = ln(x) + ln(y), it's
not ln(xy) = ln(x)
* ln(y) as you've written.
I'll re-do it for you. Let's begin:
y = \sqrt{x} e ^ {x ^ 2} (x ^ 2 + 1) ^ {10}
Take logs of
both sides yields:
\Rightarrow \ln( y ) = \ln \left( \sqrt{x} e ^ {x ^ 2} (x ^ 2 + 1) ^ {10} \right)
Use the formula: ln(xy) = ln(x) + ln(y), and ln(a
b) = b ln(a) to expand all the terms on the RHS:
\Rightarrow \ln( y ) = \frac{1}{2} \ln (x) + x ^ 2 \ln (e) + 10 \ln (x ^ 2 + 1)
ln(e) = 1, so, let's simplify it a bit:
\Rightarrow \ln( y ) = \frac{1}{2} \ln (x) + x ^ 2 + 10 \ln (x ^ 2 + 1)
If the two functions (say, f(
x), and g(
x)) are the same (f(
x) = g(
x), for all
x), then so are their derivatives (i.e, f'(
x) = g'(
x), for all
x).
So, taking the derivatives of
both sides with respect to x (not just
one side as you did), we have:
\Rightarrow \frac{y'_x}{y} = \left( \frac{1}{2} \ln (x) + x ^ 2 + 10 \ln (x ^ 2 + 1) \right)'_x
Now, what you have to do is to take the derivatives of the RHS, then multiply y over, and arrive at the desired answer.
Hope that you can go from here. :)
--------------------------------
Btw, you should put a '\' in front of a function in LaTeX to make it looks
un-italics.
Compare the two:
\ln x returns: \ln x
whereas ln x returns: ln x
The first one looks somewhat better, eh? :)
And to insert texts in LaTeX, we use the function \mbox{
your text here}.
\mbox{LaTeX is wonderful :D} is way better than LaTeX is wonderful :D. See? :)
Hint: Click on LaTeX images to see ther codes..