Calculate the derivative of the following functions.
y = (p+3)² sin p²

Find an equation of the line tangent to the following curves at the given value of x.

y = 1+2 sin x; x = π/6

The right-sided and left-sided derivatives of a function at a point $a$ are given by $f_{+}^{\prime }\left(a\right)=\underset{h\to 0^{+}}{\lim }\frac{f(a+h)-f\left(a\right)}{h}$ and $f_{-}^{\prime }\left(a\right)=\underset{h\to 0^{-}}{\lim }\frac{f(a+h)-f\left(a\right)}{h}$, respectively, provided these limits exist. The derivative $f^{\prime }\left(a\right)$ exists if and only if $f_{+}^{\prime }\left(a\right)=f_{-}^{\prime }\left(a\right)$.

Compute $f_{+}^{\prime }\left(a\right)$ and $f_{-}^{\prime }\left(a\right)$ at the given point $a$.

$f\left(x\right)=\mid x-2\mid$; $a=2$

Tangent lines Suppose f(2)=2 and f′(2) =3. Let g(x) = x²f(x) and h(x) = f(x) / x−3.

b. Find an equation of the line tangent to y = h(x) at x=2.

The right-sided and left-sided derivatives of a function at a point $a$ are given by $f_{+}^{\prime }\left(a\right)=\underset{h\to 0^{+}}{\lim }\frac{f(a+h)-f\left(a\right)}{h}$ and $f_{-}^{\prime }\left(a\right)=\underset{h\to 0^{-}}{\lim }\frac{f(a+h)-f\left(a\right)}{h}$, respectively, provided these limits exist. The derivative $f^{\(\prime\)}\(\left\)(a\(\right\))$ exists if and only if $f_{+}^{\(\prime\)}\(\left\)(a\(\right\))=f_{-}^{\(\prime\)}\(\left\)(a\(\right\))$.

Compute $f_{+}^{\(\prime\)}\(\left\)(a\(\right\))$ and $f_{-}^{\(\prime\)}\(\left\)(a\(\right\))$ at the given point $a$.

$f\left(x\right)=\{\begin{array}{c}4-x^2\text{ if }x\le 1\\ 2x+1\text{ if }x>1\end{array}$; $a=1$

First and second derivatives Find f′(x),f′′(x).

f(x) = x/x+2

Suppose f(3) = 1 and f′(3) = 4. Let g(x) = x² + f(x) and h(x) = 3f(x).

Find an equation of the line tangent to y = g(x) at x = 3.