Logarithmic and Trigonometric Bifunctional Derivative

Calculus Level 3

Given that $f(x) = \ln (\cos(x) \tan(x) )$ . It is defined on the subset of reals where the expression is valid.

Find the first derivative of $f(x)$ i.e., $f'(x)$ at $x = \dfrac{\pi }{4}$ .

0

\infty

Undefined at the given

x

value

-\infty

1

-1

1 solution

David Hontz
May 13, 2016

First: tan(x) = $\frac{sin(x)}{cos(x)}$ ; therefore cos(x)tan(x) = $\frac{cos(x)sin(x)}{cos(x)}$ = sin(x)

Now f(x) = ln[sin(x)]

f'(x) = $\frac{1}{sin(x)}$ $\frac{cos(x)}{1}$ = $\frac{cos(x)}{sin(x)}$ = cot(x)

Finally, plug in $\frac{π}{4}$ into f'(x)

f'( $\frac{π}{4}$ ) = 1

Chain Rule can also be followed without simplifying but this is shorter than that.

Good!

Samara Simha Reddy - 5 years, 1 month ago

Thank you.

David Hontz - 5 years, 1 month ago