complicated logarithmic differentiation

• Feb 20th 2009, 12:48 AM
katieeej
complicated logarithmic differentiation
i'm so sorry for keep asking questions but im really confused for this one.
the problem is y=(e^(x-1)*sin^2*x)/(((x^2)+5)^2x)

the trig function (sin^2 + 5), i don't know how to get a derivative out of that.

help me plz!
• Feb 20th 2009, 02:51 AM
tom@ballooncalculus
Hi katieeej -

if you mean $\sin^2 x + 5$ then this calls for a simple application of the chain rule for differentiation...

http://www.ballooncalculus.org/asy/diffChain/sinSqd.png

...where straight continuous line differentiate downwards with respect to x and the straight dashed line with respect to the dashed balloon expression, so that the triangular network satisfies the chain rule...

http://www.ballooncalculus.org/chain_rule.png

... though I'm not sure you meant \sin^2 x...

Don't integrate - balloontegrate!

Balloon Calculus: worked examples from past papers
• Feb 20th 2009, 02:59 AM
katieeej
clarification
the equation is

y=

e^(x-1)*sin^2(x)
-------------------- division
(x^2 +5)^2x

i started off like
ln y = ln(e^(x-1)sin^2 (x)) - ln (x^2 +5)^2x
= ln(e^(x-1))+ln (sin^2 (x))- 2x ln (x^2 +5)
= x-1 ln e +ln sin^2 (x) - 2x ln (x^2+5)
=x-1+ln (sin^2 (x)) -2x ln (x^2 +5)

then i don't know how to get derivatives.
• Feb 20th 2009, 03:46 AM
tom@ballooncalculus
• Feb 20th 2009, 04:05 AM
tom@ballooncalculus
Then the product rule...

And then you need to apply the chain rule to $\ln | \sin^2 x |$...
And using the derivative of $\sin^2 x$ which is the same as that of $\sin^2 x + 5$ in the first picture...