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.

{Use of Tech} The Witch of Agnesi The graph of y = a³ / (x² + a²), where a is a constant, is called the witch of Agnesi (named after the 18th-century Italian mathematician Maria Agnesi).

Let a = 3 and find an equation of the line tangent to y = 27 / (x² + 9) at x = 2.

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

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 $a$.

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

Calculate the derivative of the following functions.

y = (p+3)² sin p²

27–76. Calculate the derivative of the following functions.

y = √f(x), where f is differentiable and nonnegative at x.