I need help with these two questions that for some reason I keep getting wrong. I am suppose to differentiate with respect to x
sin((x^2+3)^4)
tan^5(x)sec^3(X)
For the last one I know the product rule needs to be used.
You need to apply chain rule to the first one:
$\displaystyle \begin{aligned}y=\sin\left[\left(x^2+3\right)^4\right]\implies \frac{\,dy}{\,dx} & = \cos\left[\left(x^2+3\right)^4\right]\cdot\frac{\,d}{\,dx}\left[\left(x^2+3\right)^4\right]\\ & = \cos\left[\left(x^2+3\right)^4\right]\cdot4\left(x^2+3\right)^3\cdot\frac{\,d}{\,dx}\le ft[x^2+3\right]\\ &= \dots\end{aligned}$
Can you finish this one?
For the second one, apply product and chain rule.
$\displaystyle \begin{aligned}
y=\tan^5x\sec^3x\implies \frac{\,dy}{\,dx} & = 5\tan^4x\cdot\frac{\,d}{\,dx}\left(\tan x\right)\cdot\sec^3x+\tan^5x\cdot3\sec^2x\cdot\fra c{\,d}{\,dx}(\sec x)\\
&=\dots\end{aligned}$
Can you finish this one?
Does this make sense?
For the first one I get
8x(x^2+3)^3 cos((x^2+3)^4)
using the chain rule (twice)
and for the second notice
tan^5(x)sec^3(x) = sin^5(x)/cos^8(x)
so differentiating using the quotient rule, I get
(5sin^4(x)cos^9(x) + 8cos^7(x)sin^6(x))/cos^16(x)
which will cancel down a bit.