With the aid of the rules for differentiation given in this section, compute [math]f^\prime = \ddxof f[/math] for each of the following functions.

• [a [math]f(x) = 3x^2 + 4x + 1[/math]
• [math]f(x) = x^2 (x + 1)[/math]
• [math]f(x) = x^3 (x + 2)^2[/math]
• [math]f(x) = (x^2 - 4)(x^2 + 2x + 3)[/math]
• [math]f(x) = 2x^2 + \frac1{3x^3}[/math]
• [math]f(x) = \frac{2x}{2-x}[/math]
• [math]f(x) = \frac{2x}{(2-x)^2}[/math]
• [math]f(x) = \frac{x^3}{x^5 + 1}[/math]
• [math]f(x) = \left( \frac{3-x}{3+x} \right)^2[/math]
• [math]f(x) = (x^2 + 1)^3[/math]
• [math]f(x) = \frac{2x+1}{x^2+x}[/math]
• [math]f(x) = (x^2 + 1)^{-1}[/math]
• [math]f(x) = (x + x^{-1})^2[/math]
• [math]f(x) = (x-a)(x-b)(x-c)[/math]

Determine an equation of the line tangent to the parabola [math]y = x^2 - 4x + 5[/math] at the point [math](1,2)[/math]. Draw the parabola and the tangent line.

The parabola [math]y = ax^2 + bx + c[/math] passes through [math](0,4)[/math] and is tangent to the line [math]2x + y = 2[/math] at the point [math](1, 0)[/math], Find the coefficients [math]a[/math], [math]b[/math], and [math]c[/math] for the parabola.

Show that if [math]f[/math], [math]g[/math], and [math]h[/math] are differentiable functions, then

What is the correct product rule for differentiation, analogous to the one in Problem Exercise, for (a) four factors, (b) [math]n[/math] factors?

Obtain an equation of the tangent line to the graph of the function [math]f(x) = \frac{x^3}{x^2 + 1}[/math] at the point where [math]x = 2[/math].

• If [math]f(z) = 2z^2 + 2 + \frac2{z^2}[/math], then [math]f^\prime (2) = \cdots[/math].
• If [math]f(z) = 2z^2 +2 + \frac2{z^2}[/math], then [math]f^\prime (x) = \cdots[/math].
• If [math]y = \frac{x+1}{x-1}[/math], then [math]\dydx = \cdots[/math].
• If [math]y = \frac1x[/math], then [math]\dydx (2) = \cdots[/math].
• If [math]f(x) = \frac{x^2 + 1}{x^2}[/math], then [math]\ddxof f (a) = \cdots[/math].
• If [math]w = 3u^2 + 4u + 2[/math], then [math]\nxder{}{w}{u} = \cdots[/math].

The parabola [math]y = ax^2 + bx + c[/math] is tangent to the line [math]y = 4x + 7[/math] at the point [math](-1, 3)[/math]. In addition, [math]\dydx (-2) = 0[/math]. Find the coefficients [math]a[/math], [math]b[/math], and [math]c[/math].

For each of the following functions [math]f[/math], compute the derivative [math]f^\prime[/math] and the second derivative [math]f^{\prime\prime}[/math]. \begin {exenum} \item[a] [math]f(x) = 3x^2 + 2x + 1[/math] \item[a] [math]f(x) = 5x + 1[/math] \item[a] [math]f(x) = \frac{x^4}{12} + \frac{x^3}6 + \frac{x^2}2 + x + 1[/math] \item[a] [math]f(t) = t^3(t^2 - 1)[/math] \item[a] [math]f(x) = x^3 + \frac1{x^2}[/math] \item[a] [math]f(s) = \frac{s^2 - 1}{s^2 + 1}[/math]. \end{itemize}

The line [math]y = 3x - 1[/math] is tangent to the graph of the function [math]f(x) = ax^3 + bx^2 + c[/math] at the point [math](1,2)[/math]. Furthermore, [math]\deriv{2}{f} (1) = 0[/math]. Compute [math]a[/math], [math]b[/math], and [math]c[/math].