Find the derivative of each:

\(f(x) = 2x\cos x - 2\sin x\)

\(h(x) = x^2\sin x\)

\(h'(x) = 2x \cdot \sin x + x^2 \cdot \cos x\)

\(h(x) = x^4e^x\)

\(h'(x) = 4x^3 \cdot e^x + x^4 \cdot e^x\)

\(f(x) = 5x^2\sin x\)

$$\begin{align}
f'(x) &= 5x^2 \cdot \cos x + \sin x \cdot 10x \\
&= 5x(x\cos x + 2\sin x)
\end{align}$$
Prof. Redden

\(\dfrac{dy}{dx}\left[f^2(0)\right]\)
$$\begin{array}{c|c|c}
x & f(x) & f'(x) \\
\hline
0 & 9 & -2 \\
1 & -3 & 1 / 5
\end{array}$$

$$\begin{align}
& \dfrac{dy}{dx}\left[f^2(0)\right] \\[1em]
={} & \dfrac{dy}{dx}\left[f(0)f(0)\right] \\[1em]
={} & f(0)f'(0) + f'(0)f(0) \\[1em]
={} & 2f(0)f'(0) \\[1em]
={} & 2(9)(-2) \\[1em]
={} & -36
\end{align}$$

\(f(x) = x^3\cos x\)

\(h(x) = 2x\sin x\)

\(g(x) = x(5x - 3x^2)\)

\(f(x) = e^{3x}\ln\left(x^2 - 5\right)\)

\(f(x) = x^2\sin x\)

\(f(x) = (5x - 9x^3)(8 + x^2)\)

\(f(x) = 4\sin x\tan x\)

\(f(x) = 5x\sin x - x^3\tan x\)

\(R(x) = (x^2 + 6)(7 - 8x)(3x - 5x^3)\)

$$\begin{array}{lll}
f(3) = 4 & g(3) = -8 \\
f'(3) = -7 & g'(3) = 5 \\
[fg]'(3) = \mathop{?}
\end{array}$$