Answer
$c=7$
Work Step by Step
Use the sum rule for the derivatives it follows:
$$f'(x)=(cx)'+(\ln(\cos(x)))'$$
$$f'(x)=c+(\ln(\cos(x)))'$$
Using the chain rule $(f(g(x)))'=g'(x)f'(g(x))$.In our case $g(x)=\cos(x)$ and $f(x)=\ln x$ so it follows:
$$f'(x)=c+(\cos(x))'\frac{1}{\cos(x)}$$
$$f'(x)=c-\sin(x)\frac{1}{\cos(x)}$$
$$f'(x)=c-\frac{\sin(x)}{\cos(x)}$$
$$f'(x)=c-\tan(x)$$
$$f'\left(\frac{\pi}{4}\right)=c-\tan\left(\frac{\pi}{4}\right)$$
$$6=c-1$$
$$6+1=c$$
$$7=c$$
