# Thread: Chain rule - what am i doing wrong?

1. ## Chain rule - what am i doing wrong?

I want to take the derivative of $\left(\frac{1976}{x}\right)^x$

Using the chain rule, i got $\left(\frac{-1976}{x}\right)\left(\frac{1976}{x}\right)^{x-1}$
But I know this is an increasing function (so the derivative should not be negative) and so I am kind of confused. Please help.

2. Originally Posted by machack
I want to take the derivative of $\left(\frac{1976}{x}\right)^x$

Using the chain rule, i got $\left(\frac{-1976}{x}\right)\left(\frac{1976}{x}\right)^{x-1}$
But I know this is an increasing function (so the derivative should not be negative) and so I am kind of confused. Please help.
Not as easy as it seems. Check here. Click on Show Steps.

3. Originally Posted by machack
I want to take the derivative of $\left(\frac{1976}{x}\right)^x$

Using the chain rule, i got $\left(\frac{-1976}{x}\right)\left(\frac{1976}{x}\right)^{x-1}$
But I know this is an increasing function (so the derivative should not be negative) and so I am kind of confused. Please help.
hmmm..

let $y = (\frac{1976}{x})^x$

$lny = xln( \frac{1976}{x} ) = xln(1976) -xlnx$

Implicitly differentiate

$\frac{1}{y} y^{ \prime} = ln(1976) - (lnx + 1)$

$y^{ \prime } = y[ln(1976) - (lnx + 1) ]$

$y^{ \prime } = (\frac{1976}{x})^x [ln(1976) - (lnx + 1) ]$

4. Ah I just realized what i did wrong (thought a^x could be differentiated into x*a^(x-1))