differential log problem

**slaypullingcat** · May 2nd 2009, 07:58 PM

Hi, this should be pretty easy for the helpers out there. Find the derivative of loge pi^x

Thank you

**Reckoner** · May 2nd 2009, 08:03 PM

Originally Posted by slaypullingcat

Hi, this should be pretty easy for the helpers out there. Find the derivative of loge pi^x

By "loge" do you mean the natural logarithm $\ln?$ In that case, use the chain rule, along with

$\frac d{dx}[\ln u]=\frac{u'}u$

and

$\frac d{dx}\left[a^x\right]=(\ln a)a^x.$

**slaypullingcat** · May 2nd 2009, 08:08 PM

thats what I ment, sorry. Can you please expand. I'm trying to teach myself this stuff. Can you show me with my questions please.

**warmonk102** · May 2nd 2009, 08:11 PM

Here it is, hopefully the extra e in the log is a mistype.

d/dx (log (pi^x))

using log definition of power to move the x

d/dx [x (log (pi) )]

then taking derivative and answer is

log(pi)

**Reckoner** · May 2nd 2009, 08:13 PM

Originally Posted by slaypullingcat

thats what I ment, sorry. Can you please expand. I'm trying to teach myself this stuff. Can you show me with my questions please.

We use the chain rule:

$\frac d{dx}\left[\ln\pi^x\right]=\frac1{\pi^x}\cdot\frac d{dx}\left[\pi^x\right]$

$=\frac{(\ln\pi)\pi^x}{\pi^x}=\ln\pi$

An even easier method is to rewrite

$\ln\pi^x=x\ln\pi$

and use the constant multiple rule.

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