Question

more chain rule

Answer 1

\[y=\sqrt{x^4+\cos(2x)}\]

Answer 2

\[g(x)=x^4+\cos(2x)=u\]
\[f(u)=\sqrt{u}\]

Answer 3

?

Answer 4

what in the...

Answer 5

First step in the chain rule differentiation:
\[\begin{align*}\frac{d}{dx}\left(x^4+\cos2x\right)^{1/2}&=\frac{1}{2}\left(x^4+\cos2x\right)^{-1/2}\frac{d}{dx}\left(x^4+\cos2x\right)
\end{align*}\]

Answer 6

Next step involves distributing the differentiating operator, then power rule for the first term and chain for the second:
\[\begin{align*}\frac{d}{dx}\left(x^4+\cos2x\right)^{1/2}&\color{lightgray}{=\frac{1}{2}\left(x^4+\cos2x\right)^{-1/2}\frac{d}{dx}\left(x^4+\cos2x\right)}\\\\
&=\frac{1}{2}\left(x^4+\cos2x\right)^{-1/2}\left(\frac{d}{dx}x^4+\frac{d}{dx}\cos 2x\right)\\\\
&=\frac{1}{2}\left(x^4+\cos2x\right)^{-1/2}\left(4x^3-\sin2x\left(\frac{d}{dx}2x\right)\right)
\end{align*}\]

Answer 7

See where this is going?

Answer 8

kind of, sec

Answer 9

the d/dx of 2x is just 2

Answer 10

yes

Answer 11

so it would be (1/2(x^4+cos2x)^-1/2)(4x^3-2sin2x)

Answer 12

yup