f(x)= 10x^(lnx)

How do you find the value of the derivative when x =8?

Printable View

- Sep 29th 2009, 07:15 PMLinaD3Derivative
f(x)= 10x^(lnx)

How do you find the value of the derivative when x =8? - Sep 29th 2009, 07:31 PMRoam
- Sep 29th 2009, 07:32 PMLinaD3
But I'm supposed to find the value of 8 for the derivative of that function. How do I take the derivative of it?

- Sep 29th 2009, 07:42 PMRoam
You have: $\displaystyle f(x) = 10x^{ln(x)}$

Apply the chain rule:

$\displaystyle

10x^{ln (x)} \frac{d}{dx} (ln^2x)

$

u=log x and n=2

$\displaystyle

= 20x^{ln(x)} ln(x) \frac{d}{dx} (ln(x))

$ and since the derivative of ln(x) = 1/x

you have: $\displaystyle f'(x) = 20 x^{ln(x)-1}ln(x)$ - Sep 29th 2009, 07:43 PMLinaD3
I don't know how to differentiate it. Can you give me a hint on how to?

- Sep 29th 2009, 07:48 PMRoam
- Sep 30th 2009, 12:52 AMmr fantastic
Your answer is correct but I cannot follow the logic.

Using logarithmic differentiation:

$\displaystyle y = 10 x^{\ln x} \Rightarrow \ln y = (\ln x)^2 + \ln 10$.

Therefore $\displaystyle \frac{1}{y} \frac{dy}{dx} = \frac{2 \ln x}{x} \Rightarrow \frac{dy}{dx} = \frac{2 y \ln x}{x} = \frac{2 \left(10 x^{\ln x}\right) \ln x}{x} = 20 x^{\ln (x) - 1} \ln x$. - Sep 30th 2009, 07:50 AMtom@ballooncalculus
Also, just in case a picture helps convey the logic that Roam (possibly) and - Wolfram|Alpha (definitely) are using...

http://www.ballooncalculus.org/asy/diffMulti/one.png

... where

http://www.ballooncalculus.org/asy/doubleChain.png

... is the chain rule for two inner functions, i.e...

$\displaystyle \frac{d}{dx}\ f(u(x), v(x)) = \frac{\partial f}{\partial u} \frac{du}{dx} + \frac{\partial f}{\partial v} \frac{dv}{dx}$

As with...

http://www.ballooncalculus.org/asy/chain.png

... the ordinary chain rule, straight continuous lines differentiate downwards (integrate up) with respect to x, and the straight dashed line similarly but with respect to the (corresponding) dashed balloon expression which is (one of) the inner function(s) of the composite expression.

_____________________________________

Don't integrate - balloontegrate!

Balloon Calculus Forum

Balloon Calculus Drawing with LaTeX and Asymptote!