Thread: Derivative of trig functions under the natural log

1. Derivative of trig functions under the natural log

Could someone please help me figure out the first and second derivative of the following function:

f(x)=ln(sec(6x)+tan(6x))

Explanation would be very appreciated.

2. Standard Chain Rule: $\dfrac{d}{dx} \ln[f(x)] = \dfrac{f'(x)}{f(x)}$

In this case you have $f'(x) = \dfrac{\frac{d}{dx} \sec(6x) + \frac{d}{dx} \tan(6x)}{\sec(6x) + \tan(6x)}$

You'll need to use the chain rule again on the numerator

3. Hello, bennettb6!

$\text{Find the first and second derivative of the following function:}$

. . $f(x)\:=\:\ln(\sec6x+\tan6x)$

Chain Rule: . $\displaystyle f'(x) \;=\;\frac{1}{\sec6x + \tan6x}\cdot(\sec6x\tan6x\cdot 6 + \sec^2\!6x\cdot 6)$

. . . . . . . . . $\displaystyle f'(x) \;=\;\frac{6\sec6x\,(\tan6x + \sec6x)}{\sec6x + \tan6x} \;=\;6\sec6x$

$\text{Then: }\;f''(x) \;=\;6\sec6x\tan6x\cdot 6 \;=\;36\sec6x\tan6x$

4. Perfect, thanks for the help!

,

derivatives of trig functions with natural logarithms

Click on a term to search for related topics.