have a few derivatives i cannot figure out any help would be greatly appreciated
1.at(a+t)^4
2.sin(2x+5)
3.ln(x cos x)
1)Product Rule:
$\displaystyle (at(a+t)^4)' = a(a+t)^4 + 4at(a+t)^3$
2)Chain rule:
Start with $\displaystyle u = 2x+5 \Rightarrow \frac{du}{dx}= 2$,
$\displaystyle (2\sin(2x+5))' =\frac{d(2\sin u)}{du} \frac{du}{dx}= (2\cos u) . 2 = 4 \cos(2x+5)$
3)Smart question, dont directly differentiate. First use log properties, make the problem easier. Then its the chain rule for the second part.
$\displaystyle (\ln(x \cos x))' = (\ln x + \ln \cos x)' = \frac{1}{x} + \frac1{\cos x}(\cos x)' = \frac{1}{x} - \tan x $