• If [math]f(x) = x^3 - x^2 +x -1[/math], then [math]\deriv{2}f = \cdots[/math].
• If [math]y = \frac{x-1}{x+1}[/math], then [math]\deriv2y = \cdots[/math].
• If [math]s = at^3 + bt^2 + ct + d[/math], where [math]a[/math], [math]b[/math], [math]c[/math], and [math]d[/math] are constants, compute [math]\nxder3st[/math].
• If [math]y = \frac1{x^2}[/math], then [math]\deriv3y (a) = \cdots[/math].

Find all the points on the graph of the function [math]\frac{x^3}3 - x^2[/math] at which the tangent line is perpendicular to the tangent line at [math](1, -\frac23)[/math].

There are many examples of a function [math]f[/math] and a number [math]a[/math] such that [math]f(a)[/math] is defined ([math]a[/math] is in the domain of [math]f[/math]) but [math]f^\prime(a)[/math] does not exist. Another way of saying the same thing is that the domain of [math]f^\prime[/math] can be a ph proper subset of the domain of [math]f[/math]. It is equally possible for [math]f^\prime (a)[/math] to be defined and [math]f^{\prime\prime} (a)[/math] not to be. Let [math]f[/math] be the function defined by

.}} </math>

• Compute [math]f^\prime[/math].
• Is [math]f[/math] a differentiable function?
• Show that [math]f^{\prime\prime}(0)[/math] does not exist, and compute [math]f^{\prime\prime}(x)[/math] for [math]x\ne 0[/math].

Same as Problem Exercise except that [math]f(x) = x^{\frac43}[/math].

• Draw the graph of the function [math]g[/math] defined by
  [[math]] g(x) = \dilemma{x^2, & x \leq 1,} {2x-1, & x \gt 1.} [[/math]]
• Compute [math]g^\prime[/math] and [math]g^{\prime\prime}[/math].
• Are [math]g[/math] and [math]g^\prime[/math] differentiable functions?