User:Alexbrewer/CalculusSandbox/1985BC6

1985BC6


The process required is to take the derivative of the function and then evaluate the result at $$x = \frac{\pi}{4}$$.

It is important to remember that you must use the quotient rule for derivatives!

Apply this rule to the function:

$$f'(x) = \frac{(1)(tanx)-(x)(\frac{1}{cos^2x})}{tan^2 x}$$

$$= \frac{tanx- \frac{x}{cos^2x}}{tan^2 x}$$

Because this is likely a calculator free question, I would stop trying to simply the equation at this point and evaluate the result at $$\frac{\pi}{4}$$.

$$= \frac{tan\frac{\pi}{4}- \frac{\frac{\pi}{4}}{cos^2\frac{\pi}{4}}}{tan^2 \frac{\pi}{4}}$$

$$= \frac{1- \frac{\frac{\pi}{4}}{\frac{1}{2}}}{1}$$

= 1 - \frac{\pi}{2} \quad = (C)