You may recall hearing about Becca and her Track and Field competition in a prior lesson. Her boyfriend had taken a picture of her just as she started to pull away from the others on the track. We learned how she might learn to identify her instantaneous speed at just the split second the picture was taken by using calculus to find a derivative.
What if, instead of just finding her speed at that split second, she wanted to find her acceleration?
Quotient Rule and Higher Derivatives
The Quotient Rule
Theorem: (The Quotient Rule) If f and g are differentiable functions at x and g(x) ≠ 0, then
In simpler notation
Keep in mind that the order of operations is important (because of the minus sign in the numerator) and \begin{align*}\left ( \frac{f}{g} \right )'\neq \frac {f'}{g'}\end{align*}. |
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Higher Derivatives
If the derivative \begin{align*}\,\! f'\end{align*} of the function \begin{align*}\,\! f\end{align*} is differentiable, then the derivative of \begin{align*}f'\end{align*}, denoted by \begin{align*}\,\! f''\end{align*} is called the second derivative of \begin{align*}\,\! f\end{align*}. We can continue the process of differentiating derivatives and obtain third, fourth, fifth and higher derivatives of \begin{align*}\,\! f\end{align*}. They are denoted by \begin{align*}\,\! f'\end{align*}, \begin{align*}\,\! f''\end{align*}, \begin{align*}\,\! f'''\end{align*}, \begin{align*}\,\! f^{(4)}\end{align*}, \begin{align*}\,\! f^{(5)}\end{align*}, . . . ,
Examples
Example 1
Earlier, you were asked how Becca could find her acceleration in addition to her speed.
Once Becca has calculated her instantaneous speed at a given point on the track by finding the derivative, she could then take the derivative of that function to find her instantaneous acceleration at the same point in the race.
By finding her instantaneous speed and acceleration at different points in the race, she can learn a lot about what points made a difference in her overall success, and also what points she needs to work on.
Example 2
Find \begin{align*}= \frac {dy}{dx}\end{align*} for \begin{align*}y = \frac {x^2-5}{x^3+2}\end{align*}.
\begin{align*}\frac {dy}{dx}\end{align*} | \begin{align*}= \frac {d}{dx}\left [ \frac{x^2-5}{x^3+2} \right ]\end{align*} | |
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\begin{align*}= \frac {(x^3+2).(x^2-5)'-(x^2-5).(x^3+2)'}{(x^3+2)^2}\end{align*} | ||
\begin{align*}= \frac {(x^3+2)(2x)-(x^2-5)(3x^2)}{(x^3+2)^2}\end{align*} | ||
\begin{align*}= \frac {2x^4+4x-3x^4+15x^2}{(x^3+2)^2}\end{align*} | ||
\begin{align*}= \frac {-x^4+15x^2+4x}{(x^3+2)^2}\end{align*} | ||
\begin{align*}= \frac {x(-x^3+15x+4)}{(x^3+2)^2}\end{align*} |
Example 3
At which point(s) does the graph of \begin{align*}y = \frac {x} {x^2+9}\end{align*} have a horizontal tangent line?
Since the slope of a horizontal line is zero, and since the derivative of a function signifies the slope of the tangent line, then taking the derivative and equating it to zero will enable us to find the points at which the slope of the tangent line is equal to zero, i.e., the locations of the horizontal tangents. Notice that we will need to use the quotient rule here:
\begin{align*}y\end{align*} | \begin{align*}= \frac{x} {x^2 + 9}\end{align*} | ||
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\begin{align*}y'\end{align*} | \begin{align*}= \frac{(x^2 + 9) \cdot f'(x) - x \cdot g' (x^2 + 9)} {(x^2 + 9)^2} = 0\end{align*} | \begin{align*}= \frac{(x^2 + 9) (1) - x (2x)} {(x^2 + 9)^2} = 0\end{align*} |
Multiply both sides by \begin{align*}(x^2 +9)^2\end{align*},
\begin{align*}x^2 + 9 - 2x^2\end{align*} | \begin{align*}= 0\end{align*} | |
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\begin{align*}x^2\end{align*} | \begin{align*}= 9\end{align*} | |
\begin{align*}x\end{align*} | \begin{align*}= \pm 3\end{align*} |
Therefore, at \begin{align*}x = -3\end{align*} and \begin{align*}x = 3\end{align*}, the tangent line is horizontal.
Example 4
Find the fifth derivative of \begin{align*}f(x) = 2x^4 - 3x^3 + 5x^2 - x - 1\end{align*}.
To find the fifth derivative, we must first find the first, second, third, and fourth derivatives.
\begin{align*}f'(x)\end{align*} | = \begin{align*}8x^3 - 9x^2 + 5x - x\end{align*} | |
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\begin{align*}f'' (x)\end{align*} | = \begin{align*}24x^2 - 18x + 5\end{align*} | |
\begin{align*}f''' (x)\end{align*} | = \begin{align*}48x - 18\end{align*} | |
\begin{align*}f^{(4)} (x)\end{align*} | = \begin{align*}48\end{align*} | |
\begin{align*}f^{(5)} (x)\end{align*} | = |
Example 5
Suppose y'(2) = 0 and (y/q)(2) = 0. Find q(2), assuming y(2) = 0.
Begin with the quotient rule:
\begin{align*}\left(\frac{y}{q}\right)' (2) = \left(\frac{y'(2)q(2) - y(2)q'(2)}{q(2)^2}\right)\end{align*} ..... Substitute
\begin{align*}(0) = \left(\frac{(0)q(2) - (0)q'(2)}{q(2)^2}\right)\end{align*} ..... Substituting again with given values
\begin{align*}0 = \left(\frac{(0)q(2)}{q(2)^2}\right)\end{align*} ..... Simplify with:\begin{align*}(0)q'(2) = 0\end{align*}
\begin{align*}0 = \frac{0}{q(2)}\end{align*}
\begin{align*}q(2) = 0\end{align*}
Example 6
Find the derivative of \begin{align*}k(x) = \frac{-2x - 4}{e^x}\end{align*}.
Use the quotient rule: Note: \begin{align*}(-2x - 4)' = -2\end{align*} and \begin{align*}(e^x)' = e^x\end{align*}
\begin{align*}\left(\frac{-2x-4}{e^x}\right)' = \frac{(-2)(e^x) - (-2x - 4)(e^x)}{e^{2x}}\end{align*} ..... Substitute
\begin{align*}\frac{2x + 2}{e^x}\end{align*} ..... Simplify
Example 7
Given \begin{align*}f(x) = (-x^4 -4x^3 - 5x^2 +3)\end{align*}. Find \begin{align*}f''(x)\end{align*} when \begin{align*}x=3\end{align*}.
Recall that \begin{align*}f''(x)\end{align*} means "The derivative of the derivative of x"
\begin{align*}f'(x) = -4x^3 -12x^2 - 10x\end{align*} ..... Use the power rule on f(x)
\begin{align*}f''(x) = -12x^2 - 24x - 10\end{align*} ..... Use the power rule on f'(x)
\begin{align*}f''(3) = -12(3)^2 -24(3) - 10 \to -108 -72 - 10 = -190\end{align*} ..... Substitute 3
\begin{align*}\therefore f''(3) = -190\end{align*}
Review
Use the quotient rule to solve:
- Suppose \begin{align*}u'(0) = 98\end{align*} and \begin{align*}(\frac{u}{q})'(0) = 7\end{align*}. Find \begin{align*}q(0)\end{align*} assuming \begin{align*}u(0) = 0\end{align*}.
- Given: \begin{align*} b(x) = \frac{x^2 - 5x + 4}{-5x + 2}\end{align*}, what is: \begin{align*} b'(2)\end{align*}?
- Given: \begin{align*}m(x) = \frac {e^x}{3x + 4}\end{align*}, what is \begin{align*} \frac{dm}{dx}\end{align*}?
- What is \begin{align*}\frac{d}{dx} \cdot \frac{sin (x)}{x - 4}\end{align*}?
- Find the derivative of \begin{align*}q(x) = \frac{x}{sin (x)}\end{align*}.
Solve these higher order derivatives:
- Given: \begin{align*}v(x) = -4x^3 + 3x^2 + 2x + 3\end{align*}, what is \begin{align*}v''(x)\end{align*}?
- Given: \begin{align*}m(x) = x^2 + 5x \end{align*}, what is \begin{align*}m''(x)\end{align*}?
- Given: \begin{align*}d(x) = 3x^4e^x\end{align*}, what is \begin{align*}d''(x)\end{align*}?
- Given: \begin{align*}t(x) = -2x^5sin (x)\end{align*}, what is \begin{align*}\frac{d^2t}{dx^2}\end{align*}?
- What is \begin{align*}\frac{d^2} {dx^2}3x^5e^x\end{align*}?
Solve:
- Find the derivative of \begin{align*}y = \frac{3} {\sqrt{x} + 3}\end{align*}.
- Find the derivative of \begin{align*}y = \frac{4x + 1} {x^2 - 9}\end{align*}.
- Newton’s Law of Universal Gravitation states that the gravitational force between two masses (say, the earth and the moon), m and M is equal two their product divided by the squared of the distance r between them. Mathematically, \begin{align*}F = G\frac{mM} {r^2}\end{align*} where G is the Universal Gravitational Constant (1.602 × 10^{-11} Nm^{2}/kg^{2}). If the distance r between the two masses is changing, find a formula for the instantaneous rate of change of F with respect to the separation distance r.
- Find \begin{align*}\frac{d} {d \psi} \left [\frac{\psi \psi_0 + \psi^3} {3 - \psi_0}\right ]\end{align*}, where \begin{align*}\psi_0\end{align*} is a constant.
- Find \begin{align*}\frac{d^3y} {dx^3} |_{x = 1},\end{align*} where \begin{align*}y = \frac{2} {x^3}\end{align*}.
Review (Answers)
To see the Review answers, open this PDF file and look for section 8.11.