Differentiation

**VonNemo19** · June 21st 2009, 09:34 AM

Differentiate

$y=(cosx)^{3x}$ with respect to x.

Using the basic differentiation rules for elementary functions, how do I approach this? What can be defined as u?

Is it $(cosx)^u$ or $u^{3x}$ ?

**apcalculus** · June 21st 2009, 10:15 AM

Originally Posted by VonNemo19

Differentiate

$y=(cosx)^{3x}$ with respect to x.

Using the basic differentiation rules for elementary functions, how do I approach this? What can be defined as u?

Is it $(cosx)^u$ or $u^{3x}$ ?

Logarithmic differentiation will do here.

$ln y = ln {(cos {x})^{3x}}$

$ln y = 3x ln {cos{x}}$

Now implicitly differentiate, then isolate $\frac{dy}{dx}$

I hope this helps.

**VonNemo19** · June 21st 2009, 10:42 AM

That's what I thought, but I've never applied logs to trig functions before, so i was hesitant. Thanx.

Difeerentiating

$\frac{d}{dx}(lny)=\frac{d}{dx}(3xlncosx)$

$\frac{y'}{y}=3x\frac{-sinx}{cosx}+3lncosx$

y=3xlncosx

$y'=9xlncosx(\frac{-xsinx}{cosx}+lncosx)$

Is this correct?

**running-gag** · June 21st 2009, 10:58 AM

Originally Posted by VonNemo19

That's what I thought, but I've never applied logs to trig functions before, so i was hesitant. Thanx.

Difeerentiating

$\frac{d}{dx}(lny)=\frac{d}{dx}(3xlncosx)$

$\frac{y'}{y}=3x\frac{-sinx}{cosx}+3lncosx$

y=3xlncosx

$y'=9xlncosx(\frac{-xsinx}{cosx}+lncosx)$

Is this correct?

No. I have marked in red what is uncorrect.

**VonNemo19** · June 21st 2009, 11:26 AM

Originally Posted by running-gag

No. I have marked in red what is uncorrect.

Duh! Unbelievable. Thanx

**Random Variable** · June 21st 2009, 11:32 AM

Let u = cos(x) and v = 3x

then differentiate $y= u^{v}$ by using the chain rule for functions of more than one variable

$\frac {dy}{dx} = \frac {\partial y}{\partial u} \ \frac {du}{dx} + \frac {\partial y}{\partial v} \ \frac {dv}{dx}$

Thread: Differentiation

LinkBack

Thread Tools

Display

Differentiation

Similar Math Help Forum Discussions

Implicit differentiation/ partial differentiation

Differentiation and partial differentiation

Differentiation

Differentiation Help

Differentiation and Implicit Differentiation

Search Tags