Find the following differentials.

• [math]\d(x^2 + x + 1) = \cdots[/math]
• [math]\d(7x + 2) = \cdots[/math]
• [math]\d(x^3+1)(5x-1)^3 = \cdots[/math]
• [math]\d\left( \frac{x-1}{x+1} \right) = \cdots[/math]
• [math]\d u^7 = \cdots[/math]
• [math]\d \left( \frac{u^2}{v^2} \right) = \cdots[/math]
• [math]\d(az^2 + bz + c) = \cdots[/math] ([math]a[/math], [math]b[/math], and [math]c[/math] are constants)
• [math]\d\sqrt{1 + \sqrt{1+x}} = \cdots[/math]
• [math]\d x = \cdots[/math]
• [math]\d (u^2 + 2)(v^3 - 1) = \cdots[/math]

If [math]y = 7x^3 + 2x + 1[/math] and [math]w = \frac1y[/math], compute the differential of the composition of [math]y[/math] with [math]w[/math]. That is, compute [math]\d w[/math] in terms of [math]x[/math] and [math]\d x[/math].

If [math]x = 16t^2 + 2t[/math] and [math]y = \frac1x[/math] and [math]z = y^2 + 1[/math], compute [math]\d z[/math] in terms of [math]t[/math] and [math]\d t[/math].

Using Leibnitz's Rule (the Product Rule), prove that [math]\d_a(fg) = f(a)\d_ag + g(a)\d_af[/math], thereby establishing rule \ref{2.6.iv}.

Using Theorem, prove that [math]\d_af^r = rf(a)^{r-1}\d_af[/math], thereby establishing rule \ref{2.6.v}.

What is the approximate change in the volume of a sphere of radius [math]10[/math] feet resulting from a change in the radius of [math]1[/math] inch?

A metal cylinder is found by measurement to be [math]3[/math] feet in diameter and [math]10[/math] feet long. What will be the error in the computed volume of the cylinder resulting from an error of

• lab{2.6.7a} [math]1[/math] inch in the diameter?
• lab{2.6.7b} [math]0.5[/math] inch in the length?
• both the errors in \ref{ex2.6.7a} and \ref{ex2.6.7b} combined?

If [math]f(x) = kx[/math], in what sense is [math]f[/math] its own differential?