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Workbook for Stewart’s "Calculus"

Cory Wilson

Section 2.3 Differentiation Formulas

Subsection 2.3.1 Before Class

https://mymedia.ou.edu/media/2.3-1/1_ra9y8bup
Figure 14. Pre-Class Video 1
https://mymedia.ou.edu/media/2.3-2/1_w84odhyp
Figure 15. Pre-Class Video 2

Subsubsection 2.3.1.1 Constant Functions

Example 2.3.1.
Use the definition of the derivative to find \(f'(x)\text{,}\) where \(f(x) = 2\text{.}\)
Solution.
0
Example 2.3.2.
Find the derivative of an arbitrary constant \(c\text{.}\)
Solution.
0

Subsubsection 2.3.1.2 Power Functions

Example 2.3.4.
Find the derivative of \(f(x) = x\)
Solution.
1
Example 2.3.5.
Find the derivative of \(f(x) = x^2\)
Solution.
\(2x\)
Example 2.3.6.
Find the derivative of \(f(x) = x^3\)
Solution.
\(3x^2\)
Example 2.3.7.
Find the derivative of \(f(x) = x^4\)
Solution.
\(4x^3\)
Question 2.3.8.
What is the pattern in these derivatives? Look at the powers...
Solution.
The powers drop to become constant multiples, and then the power gets reduced by one.
Example 2.3.10.
Find \(\dfrac{d}{dx}[x^{99}]\)
Solution.
\(99x^{98}\)

Subsubsection 2.3.1.3 New Derivatives from Old

Example 2.3.11.
Using definition of the derivative, find the derivative of \(f(x) = 3x^2\)
Solution.
\(6x\)
Example 2.3.12.
Using definition of the derivative, find the derivative of \(f(x) = -2x^3\)
Solution.
\(-6x^2\)
Question 2.3.13.
What relationship do you notice between these derivatives and the coefficients?
Solution.
When there’s a coefficient on a function, the derivative incorporates the coefficient
There are three more important rules:
Question 2.3.16.
Restate the sum/difference rule in words.
Solution.
The derivative of a sum/difference is the sum/difference of derivatives
Example 2.3.18.
Compute the derivative of \(f(x) = x^3\cdot x^5\)
Solution.
\(f'(x) = 3x^2\cdot x^5 + 5x^4\cdot x^3 = 8x^7\)
Example 2.3.20.
Compute the derivative of \(g(x) = \dfrac{x}{x+1}\)
Solution.
\(g'(x) = \dfrac{1}{(x+1)^2}\)

Subsection 2.3.2 Pre-Class Activities

Example 2.3.21.

Find the derivative of the function \(f(x) = 2^{40}\)
Solution.
\(f'(x) = 0\)

Example 2.3.22.

Find the derivative of the function \(f(t) = 1.4t^5-2.5t^2+6.7\)
Solution.
\(f'(t) = 7t^4-5t\)

Example 2.3.23.

Find the derivative of the function \(B(y) = cy^{-6}\text{,}\) where \(c\) is an arbitrary constant
Solution.
\(B'(y) = -6cy^{-5}\)

Example 2.3.24.

Find the derivative of the function \(y = \dfrac{x^2 + 4x+3}{\sqrt{x}}\)
Solution.
\(\dfrac{dy}{dx} = \dfrac{3}{2}x^{1/2} + 2x^{-1/2} - \dfrac{3}{2}x^{-3/2}\)

Subsection 2.3.3 In Class

Example 2.3.25.

For the following functions, find the derivatives. Then, find an equation for the tangent line to the function at the given point.
  1. \(S(p) = \sqrt{p}-p\) at \((9,-6)\)
  2. \(F(r) = \dfrac{5}{r^3}\) at \((-2,-5/8)\)
  3. \(A(r) = \pi r^2\) at \((1,\pi)\)
Solution.
  1. \(S'(p) = \dfrac{1}{2}p^{-1/2} - 1\) and the tangent line \(y=-\dfrac{5}{6}x + \dfrac{3}{2}\)
  2. \(F'(r) = -15r^{-4}\) and the tangent line is \(y=-\dfrac{15}{16}x -\dfrac{5}{2}\)
  3. \(A'(r) = 2\pi r\) and the tangent line is \(y=2\pi x -\pi\)

Example 2.3.26.

Find the derivative of the following functions.
  1. \(\displaystyle F(y) = \lrpar{\dfrac{1}{y^2}-\dfrac{3}{y^4}}(y+5y^3)\)
  2. \(\displaystyle J(v) = (v^3-2v)(v^{-4}+v^{-2})\)
  3. \(\displaystyle f(x) = \dfrac{1+2x}{3-4x}\)
  4. \(\displaystyle Y(t)=\dfrac{1}{t^3+2t^2-1}\)
  5. \(\displaystyle h(u) = \dfrac{(u+2)^2}{1-u}\)
  6. \(\displaystyle f(t) = \dfrac{\sqrt[3]{t}}{t-3}\)
Solution.
  1. \(\displaystyle F'(y) = (-2y^{-3}+12y^{-5})(y+5y^3) + (y^{-2}-3y^{-4})(1+15y^2)\)
  2. \(\displaystyle J'(v) = (3v^2-2)(v^{-4}+v^{-2})+(v^3-2v)(-4v^{-5}-2v^{-3})\)
  3. \(\displaystyle f'(x) = \dfrac{(3-4x)(2)-(1+2x)(-4)}{3-4x}\)
  4. \(\displaystyle Y'(t)=-\dfrac{-3t^2-4t}{(t^3+2t^2-1)^2}\)
  5. \(\displaystyle h'(u) = \dfrac{(2u+4)(1-4u)-(-1)(u^2+4u+4)}{(1-u)^2}\)
  6. \(\displaystyle f'(t) = \dfrac{\lrpar{\dfrac{1}{3}t^{-2/3}}(t-3)-(\sqrt[3]{t})(1)}{(t-3)^2}\)

Example 2.3.27.

Let \(f(x) = 2x^3-x^2+1\text{.}\) The normal line through a point is the line which is perpendicular to the tangent line.
  1. Find the equation of the tangent line to \(f(x)\) at \((1,3)\text{.}\)
  2. Find the equation of the normal line to \(f(x)\) at \((1,3)\text{.}\)
Solution.
  1. \(\displaystyle y=4x-1\)
  2. \(\displaystyle y=-\dfrac{1}{4}x+\dfrac{13}{4}\)

Example 2.3.28.

Let \(f(x) = \dfrac{2x}{x+1}\text{.}\) Find the equation of the tangent line to \(f(x)\) at \((1,1)\text{.}\)
Solution.
\(y=x\)

Example 2.3.29.

If \(H(x) = xf(x)\text{,}\) find an expression for \(H'(x)\)
Solution.
\(H'(x) = f(x) + xf'(x)\)

Example 2.3.30.

Suppose that \(f(5) = 1\text{,}\) \(f'(5) = 6\text{,}\) \(g(5) = -3\text{,}\) and \(g'(5) = 2\text{.}\) Find
  1. \(\displaystyle (fg)'(5)\)
  2. \(\displaystyle (f/g)'(5)\)
  3. \(\displaystyle (g/f)'(5)\)
Solution.
  1. \(\displaystyle -16\)
  2. \(\displaystyle -\dfrac{20}{9}\)
  3. 20

Example 2.3.31.

If \(h(2) = 4\) and \(h'(2) = -3\text{,}\) find \(\dfrac{d}{dx} \lrpar{\dfrac{h(x)}{x^2}}\bigg\rvert_{x = 2}\)
Solution.
\(\dfrac{d}{dx} \lrpar{\dfrac{h(x)}{x^2}}\bigg\rvert_{x = 2}=-\dfrac{7}{4}\)

Example 2.3.32.

If \(f\) and \(g\) are the functions on the graph shown, let \(u(x) = f(x)g(x)\) and \(v(x) = f(x)/g(x)\text{.}\) Find \(u'(1)\) and \(v'(5)\text{.}\)
Solution.
\(u'(1) = 0\) and \(v'(5) = -\dfrac{2}{3}\)

Subsection 2.3.4 After Class Activities

Example 2.3.33.

Suppose that \(f(4) = 2\text{,}\) \(g(4) = 5\text{,}\) \(f'(4) = 6\text{,}\) and \(g'(4) = -3\text{.}\) Find \(h'(4)\) if:
  1. \(\displaystyle h(x) = 3f(x) + 8g(x)\)
  2. \(\displaystyle h(x) = f(x)g(x)\)
  3. \(\displaystyle h(x) = \dfrac{f(x)}{g(x)}\)
  4. \(\displaystyle h(x) = \dfrac{g(x)}{f(x) + g(x)}\)
Solution.
  1. \(\displaystyle -6\)
  2. \(\displaystyle 24\)
  3. \(\displaystyle \dfrac{36}{25}\)
  4. \(\displaystyle -\dfrac{36}{49}\)

Example 2.3.34.

If \(f\) is a differentiable function, find an expression for the derivative of each of the following:
  1. \(\displaystyle y = x^3f(x)\)
  2. \(\displaystyle y = \dfrac{f(x)}{x}\)
  3. \(\displaystyle y = \dfrac{x^2}{f(x)}\)
  4. \(\displaystyle y = \dfrac{1 + xf(x)}{\sqrt{x}}\)
Solution.
  1. \(\displaystyle \dfrac{dy}{dx} = 3x^2f(x) + x^3f'(x)\)
  2. \(\displaystyle \dfrac{dy}{dx} = \dfrac{xf'(x)-f(x)}{x^2}\)
  3. \(\displaystyle \dfrac{dy}{dx} = \dfrac{2xf(x)-x^2f'(x)}{[f(x)]^2}\)
  4. \(\displaystyle \dfrac{dy}{dx} = -\dfrac{1}{2}x^{-3/2} + \dfrac{1}{2}x^{-1/2}f(x) + x^{1/2}f'(x)\)

Example 2.3.35.

For what values of \(x\) does the graph of \(f(x) = x^3 + 3x^2 + x + 3\) have a horizontal tangent?
Solution.
\(x = \dfrac{-6\pm \sqrt{24}}{6}\)

Example 2.3.36.

Show that the vertex of any quadratic function occurs at \(x=-\dfrac{b}{2a}\)
Solution.
A general quadratic is given by \(f(x) = ax^2+bx+c\text{;}\) the vertex has a horizontal tangent line, so its derivative must be zero. Then, \(f'(x) = 2ax + b\) so that if \(f'(x) = 0\) then we have \(2ax+b = 0\iff x = -\dfrac{b}{2a}\text{.}\)
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