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!

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- Feb 20th 2009, 01:02 AMkatieeejlogarithmic 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, 01:10 AMtah
- Feb 20th 2009, 01:15 AMkatieeej
the equation is

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

its really complicated i'm sorry!

it's sin square of (x) and (x^2 +5) is x square and plus 5, separately.

thanks! - Feb 20th 2009, 01:44 AMsynclastica_86
We have:

$\displaystyle y=\frac{e^{x-1}\sin^2{(x)}}{(x^2+5)^{2x}}={e^{x-1}\sin^2{(x)}}{(x^2+5)^{-2x}}$

To make things cleaner, we use the product rule and let:

$\displaystyle f(x)=e^{x-1}\sin^2(x)$

and

$\displaystyle g(x)=(x^2+5)^{-2x}=e^{\ln(x^2+5)^{-2x}}=e^{-2x\ln(x^2+5)}$

We now have:

$\displaystyle \frac{dy}{dx}=f'(x)g(x)+g'(x)f(x)$

Using the product:

$\displaystyle f'(x)=e^{x-1}\sin^2(x)+2\sin(x)\cos(x)e^{x-1}=e^{x-1}\sin(x)(2\cos(x)+\sin(x))$

Using the chain rule:

$\displaystyle g'(x)=e^{-2x\ln(x^2+5)}\frac{d}{dx}-2x\ln(x^2+5)$

$\displaystyle =-2(x^2+5)^{-2x}\frac{d}{dx}x\ln(x^2+5)$

$\displaystyle =-2(x^2+5)^{-2x}(x\frac{d}{dx}\ln(x^2+5)+\ln(x^2+5))$

$\displaystyle =-2(x^2+5)^{-2x}(\frac{x}{x^2+5}(2x)+\ln(x^2+5))$ here I use the property: $\displaystyle \frac{d}{du}\ln(u)=\frac{1}{u}$

$\displaystyle =-2(x^2+5)^{-2x}(\frac{2x^2}{x^2+5}+\ln(x^2+5))$

Now I'll leave it for you to plug in and simplify. - Feb 20th 2009, 01:49 AMtah
the derivative looks a bit ugly but it follows from the formulas

$\displaystyle (uv)' = u'v + uv'\ and\ (\frac{u}{v})' = \frac{u'v-uv'}{v^2}$

which I am sure you have already used.

Try it out and compare your result to

$\displaystyle \frac{e^{x-1}sin(x) (x^3 sin(x) - 5 x^2sin(x) + 5 x sin(x) - 5 sin(x) + 2 x^3cos(x) + 10 x cos(x))}{x ^2(x ^2+ 5)^3 }$

which was done by a symbolic manipulation software (Yes I didn't do it :)) - Feb 20th 2009, 01:59 AMGrandadDifferentiating
Hello katieeejI'm sure you'll find it easier to begin by taking logs:

$\displaystyle ln(y) = (x-1) + 2ln(\sin x) - 2x \cdot ln(x^2+5)$

*Then*differentiate:

$\displaystyle \frac{1}{y}\frac{dy}{dx} = ...$ etc.

and finally multiply both sides by $\displaystyle y$ (= ...original function of $\displaystyle x$).

Grandad