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Section Derivatives of Polynomial, Root, and Rational Functions

Problem 2.2.

Compute the derivative of each function using Definition 2.1.

  1. \(f(x) = x^2 - 3x\)

  2. \(g(x) = 3x-7\)

  3. \(h(t) = 4t^3\)

  4. \(\dsp u(t) = \frac{1}{t}\)

  5. \(\dsp v(x) = \frac{x-5}{x+2}\)

Problem 2.3.

Compute the following using the derivatives you computed in the previous problem. Graph both the original function and the tangent line at the point of interest.

  1. \(f'(x)\) at \(x=5\)

  2. \(h'(t)\) at \(t=e\) (Recall that Euler's constant, \(e\text{,}\) is approximately \(2.71828...\))

  3. \(v'(x)\) at \(x=-2\)

Algebra Reminder. An expression such as \(\dsp \frac{\sqrt{a}-\sqrt{b}}{a-b},\) can be simplified by multiplying the numerator and the denominator of the fraction by \(\dsp \sqrt{a}+\sqrt{b}.\) This is called rationalizing the numerator and \(\dsp \sqrt{a}+\sqrt{b}\) is called the conjugate of \(\dsp \sqrt{a}-\sqrt{b}.\)

Problem 2.4.

If \(f(x) = \sqrt{x+1}\text{,}\) compute \(f'\text{.}\) Graph \(f\) and the tangent line to \(f\) at \((1,f(1))\) and state the slope of the tangent line.

Problem 2.5.

If \(f(x) = 2x^3 + 3x^2 - 7\text{,}\) compute \(f'\text{.}\) Graph \(f\) and compute the equation of the tangent line to \(f\) at \((1,f(1)).\)

Problem 2.6.

If \(\dsp f(x) = \frac{1}{x+1}\text{,}\) compute \(f'.\)

Problem 2.7.

If \(f(x) = \sqrt{x+1}\text{,}\) find the equation of the tangent line at \(x = a.\)

Problem 2.8.

For each of the following, check the left- and right-hand limits to determine if the limit exists:

  1. \(\dsp \lim_{x \rightarrow 0}\frac{0}{x}\)

  2. \(\dsp \lim_{x \rightarrow 0}\frac{|x|}{x}\)

Definition 2.9.

We say that a function, f, is differentiable at a if \(f'(a)\) exists.

Definition 2.10.

We say that a function, f, is differentiable if \(f\) is differentiable at every point in its domain.

Definition 2.11.

We say that a function, f, is left differentiable at a if \(\dsp \lim_{t \rightarrow a^-} \frac{f(t) - f(a)}{t-a}\) exists. If \(f\) is left differentiable at \(a\text{,}\) we denote the left derivative of \(f\) at \(a\) by \(f'(a-).\) Right differentiability is defined similarly.

Definition 2.12.

We say that \(f\) is differentiable on (a,b) if \(f\) is differentiable at \(t\) for all \(t\) in \((a,b)\text{.}\)

Definition 2.13.

We say that f is differentiable on [a,b] if f is differentiable on (a,b) and \(f'(a+)\) and \(f'(b-)\) both exist.

Problem 2.14.

Let \(f(x) = |x - 2|.\)

  1. Graph \(f.\)

  2. Determine the equation of the line tangent to \(f\) at \((-1,3).\)

  3. Determine the equation of the line tangent to \(f\) at \((5,3).\)

  4. Compute \(f'(2+)\) and \(f'(2-).\)

  5. Does \(f'(2)\) exist? Why?

  6. What is the derivative of \(f?\)

Example 2.15.

Illustrate ways in which a function might fail to be differentiable at a point \(c\) such as:

  • the point \(c\) might not be in the domain of the function,

  • the function might have a vertical tangent line at \(c\text{,}\) or

  • the left and right-hand derivatives might not agree at \(c\) (i.e. \(f'(c+) \neq f'(c-)\)).

Problem 2.16.

For each of the following, use a sketch of the graph to determine the set of those points at which the function is not differentiable.

  1. \(\dsp f(x) = \frac{3}{1-x^{2}}\)

  2. \(g(x) =\sqrt{5+x}\)

  3. \(i(x) = \sec(x)\)

We know how to compute derivatives of functions using limits. By proving theorems we will increase the ease with which we compute derivatives of more challenging functions.

Problem 2.18.

Compute \(f'\) for each function listed, using the definition of the derivative.

  1. \(f(x) = a,\) where \(a\) is a constant

  2. \(f(x) = x\)

  3. \(f(x) = x^2\)

  4. \(f(x) = x^3\)

  5. \(f(x) = x^4\)

At this point we have used the definition of the derivative (limits) to compute the derivatives of many different functions, including: \(a\) (constant function), \(x\) (the identity function), \(x^2\text{,}\) \(x^3\text{,}\) \(4x^3\text{,}\) \(x^4\text{,}\) \(\dsp \sqrt{x+1}\text{,}\) \(2x^3 + 3x^2 - 7\text{,}\) \(\dsp \frac{1}{x}\text{,}\) \(\dsp \frac{1}{x+1}\text{,}\) \(\dsp \frac{x-5}{x+2}\text{,}\) \(|x-2|\text{,}\) and \(x^3 + 2\text{.}\) Based on the derivatives we have computed thus far, you can likely guess the answer to the next question.

Question. What is the derivative of \(f(x) = x^{200}\text{?}\) Guess. \(f'(x)=200x^{199}\text{.}\)

How do we know if our guess is correct? We don't. We must show that it is true using the definition of the derivative. Rather than work this one problem and still not know what the derivative of \(g(x) = x^{201}\) is, we'll let \(n\) represent some unknown natural number and compute the derivative of \(f(x) = x^n.\) Because we will not choose a specific value for \(n,\) we will now know the derivative of an entire class of functions.

Problem 2.19.

Power Rule.

Let \(n\) be a natural number and use Definition 2.1 to show that the derivative of \(f(x) = x^n\) is \(f'(x)=nx^{n-1}\text{.}\)

Problem 2.20.

Constant Rule. Show that if \(k\) is a constant (i.e. any real number) and \(h(x) = kf(x)\text{,}\) then \(h'(x) = kf'(x)\text{.}\)

Note. The previous problem is not stated accurately. Here is a precise statement. Show that if \(k\) is a constant (i.e. any real number) and \(f\) is differentiable on \((a,b)\) and \(h(x) = kf(x)\) for all \(x\) in the interval \((a,b)\text{,}\) then \(h'(x) = kf'(x)\) for all \(x\) in \((a,b)\text{.}\)

Problem 2.21.

Sum Rule. Show that if each of \(f\) and \(g\) is a function and \(S(x) = f(x) + g(x)\text{,}\) then \(S'(x) = f'(x) + g'(x)\text{.}\)

Problem 2.22.

Difference Rule.

Show that if each of \(f\) and \(g\) is a function and \(D(x) = f(x) - g(x)\text{,}\) then \(D'(x) = f'(x) - g'(x)\text{.}\)

The next theorem allows us to avoid using limits to determine if a function is continuous, since it states that all differentiable functions are also continuous functions.

Problem 2.25.

Product Rule. Show that if \(P(x) = f(x)g(x)\text{,}\) then \(P'(x) = f'(x)g(x) + f(x)g'(x)\text{.}\)

Problem 2.26.

Quotient Rule.

Show that if \(\dsp Q(x) = \frac{f(x)}{g(x)}\text{,}\) then \(\dsp Q'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}.\)

Problem 2.19 only proved the power rule for exponents that were positive integers. With the new tools, the next problem extends the power rule to include negative and zero exponents. A natural question would then be, “Is this true for exponents that are real numbers but not integers?” The answer is “yes” but we'll simply assume this and not prove it in this course.

Problem 2.27.

Power Rule Revisited. Show that if \(n\) is any integer and \(f(x) =x^n\text{,}\) then \(f'(x) = nx^{n-1}\text{.}\) Before you start, ask yourself how this is different from Problem 2.19.

Problem 2.28.

Suppose \(a,b,c,\) and \(d\) are constants such that \(d \neq 0\text{,}\) and find \(f'\) where \(\dsp f(x) = \frac{ax^2 + bx + c}{d}.\)

Problem 2.29.

Find \(f'\) when \(f\) is the function defined by \(\displaystyle{f(x) = (x - \frac{1}{x})(2x^2+3x+4)}.\)

Problem 2.30.

Find \(f'\) when \(f\) is the function defined by \(\displaystyle{f(x) = \frac{x-1}{x+1} \cdot \frac{x-2}{x+2}}.\)

Problem 2.31.

Find \(f'\) when \(f\) is the function defined by \(\displaystyle{f(r) = \frac{r^2s^2-r^4}{s-m}}\) and \(s\) and \(m\) are real numbers and \(s \neq m\text{.}\)

Since the derivative of a function is itself a function, there is no reason not to take the derivative of the derivative. When this is done, the new function is called the second derivative and is denoted by \(f''\text{.}\) Similarly the third derivative is denoted by \(f'''\text{.}\)

Problem 2.32.

Find \(f''\) when \(\displaystyle{f(x) = \frac{2}{x} - \frac{x^4}{2}}\text{.}\)

You have solved algebraic equations such as

\begin{equation*} 5x + 2 = 3x + 4 \;\;\; \mbox{or} \;\;\; 3x^2 - 2x + 1 = 0 \end{equation*}

where the goal is to find each value of \(x\) that solves the equations. Engineering applications often need to solve differential equations such as

\begin{equation*} f'(x) = x f(x) \;\;\; \mbox{or} \;\;\; f''(x) = x^3 \;\;\; \mbox{or} \;\;\; f''(x) + f(x) = 0 \end{equation*}

where the goal is to find each function \(f\) that satisfies the equation. In the real world, solutions to differential equations might represent the shape of an airplane wing or the temperature distribution of an engine block.

Problem 2.33.

Show that if \(x\) is not 0, then the function \(\displaystyle{f(x) = \frac{2}{x}}\) is a solution to the equation \(x^3f''(x) + x^2f'(x) - xf(x) = 0\text{.}\)

Problem 2.34.

Find one function \(f\) that satisfies \(f'(x) = 4x^2 - 2x.\)

We have been very careful up to this point in the book to write the equations in functional notation. That is, we never wrote, \(y=x^2,\) but rather wrote \(f(x) = x^2\) to emphasize that \(f\) is the function and it depends on the independent variable, \(x.\) As you will see in other courses, it is very common to write, \(y=x^2\) and then follow this with \(y'=2x.\) Differential equations are almost always written in this short-hand, omitting the independent variable where possible, and from this point forward, we too will use this short-hand on occasion.

Problem 2.35.

Sketch the graph of \(y = 2x^3 - 3x^2 - 12x + 2\) and locate (by using the derivative) the points on the graph that have a horizontal tangent line.

Problem 2.36.

Find an equation for the line tangent to \(\displaystyle{y = \frac{2-x}{2+x}}\) at \((1,\frac{1}{3})\text{.}\)

Problem 2.37.

Let \(\dsp f(x) = \frac{2}{x}\) and \(c\) be any real number. Find the area of the triangle (\(\dsp A=\frac{1}{2}bh\)) formed by the tangent line to the graph at \((c,f(c))\) and the coordinate axes.

Problem 2.38.

Show that if the function f is differentiable at x and \(g(x) = (f(x))^2\text{,}\) then \(g'(x) = 2f(x)f'(x)\text{.}\)

Problem 2.39.

Sketch a graph of

\begin{equation*} f(x) = \left\{ \begin{array}{ll} x^2 \amp \mbox{if } x \leq 1 \\ \sqrt{x} \amp \mbox{if } x > 1. \end{array} \right. \end{equation*}

Use left and right hand derivatives at \(x=1\) to determine if \(f\) is differentiable at \(x=1.\)

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