Short and Sweet Calculus

3.1 The Concept of a Derivative

3.1.1 The Tangent Problem

It had been a long-lasting problem in mathematics how to define the tangent line to an arbitrary curve. The word “tangent” originates from the Latin verb “tangere”, which means “to touch.” So, we intuitively know that the tangent line to a curve at a point is a straight line that touches the curve at that point.

For a circle, the problem is easy to solve. A tangent line to a curve is a line that meets the circle at one point. Other lines intersect the circle at two different points or completely miss it (Figure 3.1(a)). The question is: can we generalize this concept and say a tangent line to a curve is line that intersects the curve at one point? Let’s look at Figure 3.1(b). We intuitively know that the line \(l\) is a tangent line at \(P\) (it touches the curve) and the line \(d\) is not, but \(l\) intersects the curve more than once and \(d\) intersects it once. So this generalization does not work.

(a) A tangent line to a circle intersects the circle at only one point (b) We intuitively know that line l is tangent to the curve at point P, but line m is not a tangent line to the curve although it intersects the curve only at one point.

In the seventeenth century, Pierre de Fermat devised a method to construct the tangent line. To draw the tangent line to a curve at a point \(P\) on it, we choose a second nearby point \(Q\) on the curve and draw the secant line \(PQ\).3 The tangent line at \(P\) is the limiting position of the secant \(PQ\) as \(Q\) moves up toward coincidence with \(P\) (Figure 3.2).

The limit of the secant slopes as Q\to P from either side is the slope of the tangent line to the curve at P and is equal to f'(x_{0})

To find the equation of the tangent line, we just need its slope because we already know that \(P\) lies on the line.4 So let’s find the slope of the tangent line.

Consider the curve \(y=f(x)\) and let \(P\left(a,b\right)\) be a fixed point on it. Because \(P\) is on the curve, we have \(b=f(a)\). Let \(Q(x,y)\) with \(y=f(x)\) be a variable point on this curve. The slope of the secant \(PQ\) is

\[m_{PQ}=\frac{\text{rise}}{\text{run}}=\frac{y-b}{x-a}=\frac{f\left(x\right)-f\left(a\right)}{x-a}.\]

Notice that we may have \(x<a\) or \(x>a\).

If we let \(x\rightarrow a\), the point \(Q\) will move along the curve and gets closer and closer to \(P\), the secant will turn about \(P\) and approach the tangent as the limiting position

The slope of the tangent line is then

\[m_{\text{tan}}=\lim_{Q\to P}m_{PQ}=\lim_{x\rightarrow a}\frac{f\left(x\right)-f\left(a\right)}{x-a}.\]

An alternative way of expressing the above limit is to use \(h\) to denote the difference

\[h=x-a.\] Then the statement \(x\rightarrow a\) is equivalent to \(h\rightarrow0\) and we can write the slope of the tangent line as

\[m_{\text{tan}}=\lim_{h\rightarrow0}\frac{f\left(a+h\right)-f\left(a\right)}{h}.\]

3.1.2 Delta Notation (Optional)

To denote the change in a variable, we may use a widely use notation, called the delta notation. In this notation, the letter \(\Delta\), which is the Greek D, is written in the front of a variable, to signify the difference between two values of the variable. Specifically if a variable \(w\) changes from the “old” value \(w_{0}\) to the “new” value \(w_{1}\), the amount of such a change is denoted by \(\Delta w\) (read “delta w”), namely \[\begin{aligned} \Delta w & =\text{change in \ensuremath{w} from the old value to the new one}\\ & =w_{1}-w_{0}.\end{aligned}\]

  • Notice that \(\Delta w\) is not the product of \(\Delta\) and \(w\), but it is a single entity called an increment of \(w\).

  • The increment can be positive or negative.

For example, consider \[y=x^{2}+3x+2.\] When \(x=2\), \(y=12\). When \(x=2.1\), \(y=12.71\). The change in \(x\) is 0.1, and the change in \(y\) is 0.71, and we write \[\Delta x=0.1,\quad\Delta y=0.71.\]

  • Using the delta notation, the slope of the secant line \(PQ\) can be written as \[m_{PQ}=\frac{\Delta y}{\Delta x},\] where \[\Delta x=x-a,\quad\text{and}\quad\Delta y=f(x)-f(a)=f(a+\Delta x)-f(a).\] The fraction \(\Delta y/\Delta x\) is commonly called the difference quotient because it is the quotient of two differences, namely \(\Delta y\) and \(\Delta x\).

  • The slope of the tangent line to \(y=f(x)\) at \(\left(a,f(a)\right)\) is thus \[\lim_{\Delta x\to0}\frac{\Delta y}{\Delta x}\quad\text{or}\quad\lim_{\Delta x\to0}\frac{f(a+\Delta x)-f(a)}{\Delta x}.\] Notice that here the symbol \(\Delta x\) is the independent variable.

3.1.3 The Derivative

The slope of the tangent to the curve \(y=f(a)\) at \(P(a,f(a))\) is called the derivative of \(f\) at \(x=a\) and is denoted by \(f'(a)\).

3.1. A function \(y=f\left(x\right)\) is said to be differentiable at \(x=a\) if

\[\lim_{h\rightarrow0}\frac{f\left(a+h\right)-f\left(a\right)}{h}\] exits. In this case, this limit is called the derivative of \(f\) at \(x=a\) and is denoted by \(f'(a)\) or \(y'(a)\).

  • We say \(f\) is differentiable on an interval \(I\), if it is differentiable at every point of the interval.

  • The process of finding \(f'(a)\) is called differentiation.

Now we can define the tangent line to a given curve \(y=f(x)\):

3.2. The tangent line to the curve \(y=f(x)\) at \(\left(a,f(a)\right)\) is the line through \(\left(a,f(a)\right)\) with slope \(f'(a)\).

Example 3.1. Find the equation of the tangent line to the curve \(y=\sqrt{x}\) at \((4,2)\).

Solution

Let \(f(x)=\sqrt{x}\). Then \[\begin{aligned} f'(4) & =\lim_{h\to0}\frac{f(4+h)-f(4)}{h}\\ & =\lim_{h\to0}\frac{\sqrt{4+h}-2}{h}\end{aligned}\] Direct substitution of \(h=0\), gives us the indeterminate form \(0/0\). To get rid of the radical in the numerator, multiply the numerator and denominator by the conjugate of the numerator \(\sqrt{4+h}+2\): \[\begin{aligned} f'(4) & =\lim_{h\to0}\left[\frac{\sqrt{4+h}-2}{h}\frac{\sqrt{4+h}+2}{\sqrt{4+h}+2}\right]\\ & =\lim_{h\to0}\frac{(\sqrt{4+h})^{2}-2^{2}}{h(\sqrt{4+h}+2)}\tag{\small recall \ensuremath{(A-B)(A+B)=A^{2}-B^{2}}}\\ & =\lim_{h\to0}\frac{\cancel{4}+h\cancel{-4}}{h(\sqrt{4+h}+2)}\tag{\small\ simplify}\\ & =\lim_{h\to0}\frac{1}{\sqrt{4+h}+2}\\ & =\frac{1}{\sqrt{4+0}+2}=\frac{1}{4}.\tag{\small substitute 0 for \ensuremath{h}}\end{aligned}\] [You could alternatively use \({\displaystyle f'(4)=\lim_{x\to2}\frac{f(x)-f(2)}{x-2}}\).]

Now that we know that the slope of the tangent line is \(m_{\text{tan}}=1/4\) and we know that it passes through \((4,2)\), we can write the equation of the tangent line as \[y-4=\frac{1}{4}(x-2)\] or \[y=\frac{1}{4}x+\frac{3}{2}.\]

3.1.4 The Derivative Function

The symbol \(f'(a)\) seems like that there is a new function \(f’\) and \(f'(a)\) is the value of this function at \(a\). In fact, for any function \(f\), its derivative \(f’\) is the new function whose value at \(x\) is given by \[\boxed{f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}.}\] Note that in the limit \(h\) is the variable and \(x\) is treated as a fixed number.

  • You may also denote the derivative of \(y=f(x)\) by \(y’\).

Example 3.2. Given \(f(x)=3x^{2}+5\), find \(f'(x)\).

Solution

\[\begin{aligned} f'(x) & =\lim_{h\to0}\frac{f(x+h)-f(x)}{h}\\ & =\lim_{h\to0}\frac{\left[3(x+h)^{2}+5\right]-\left[3x^{2}+5\right]}{h}\\ & =\lim_{h\to0}\frac{\left[3(x^{2}+2xh+h^{2})+5\right]-\left[3x^{2}+5\right]}{h}\tag{\small expand}\\ & =\lim_{h\to0}\frac{\cancel{3x^{2}}+6xh+3h^{2}+\bcancel{5}\cancel{-3x^{2}}\bcancel{-5}}{h}\tag{\small simplify}\\ & =\lim_{h\to0}(6x+3h)=6x.\end{aligned}\] To find \(f(x+h)\), replace \(x+h\) for \(x\) in the formula of \(f(x)\).

3.1.5 The Leibniz Notation

Using the delta notation, we can write \[f'(x)=\lim_{\Delta x\to0}\frac{\Delta y}{\Delta x}.\] Leibniz denoted the derivative in the form \(dy/dx\) (read “dy over dx”). In this notation, the definition of the derivative is \[\frac{dy}{dx}=\lim_{\Delta x\to0}\frac{\Delta y}{\Delta x}.\] At present, we take the symbol \(dy/dx\) not as a fraction, but as one undivided symbol to represent the derivative. Later, we will consider what meaning may be given to \(dy\) and \(dx\), separately. At this stage, the form suggests simply the fraction, which has approached a definite limiting value.

Slightly different forms of the symbol \(dy/dx\) are \[\boxed{\frac{df}{dx},\qquad,\frac{df(x)}{dx},\qquad\text{and}\quad\frac{d}{dx}f(x).}\] In the last one, \(d/dx\) can be considered as differentiating operator, which indicates that any function that follows it is to be differentiated with respect to \(x\).

The advantage is that it clearly indicates the the independent variable. The advantage of the Leibniz notation for derivatives is that this notation helps us remember some differentiation rule (specifically the chain rule) easier.

However, the Leibniz notation is not perfect. If we want to write that the derivative of \(f\) at \(x=2\) is 5, the notation \(f'(x)\) is superior. We can simply substitute 2 for \(x\) and write \(f'(2)=5\). In this case, if we wish to use the notation \(dy/dx\), we would have to write \[\boxed{\frac{dy}{dx}(2)=5,\qquad\left.\frac{dy}{dx}\right|_{x=2}=5,\qquad\text{or}\qquad\left(\frac{dy}{dx}\right)_{x=2}=5,}\] In the above notations, we may replace \(y\) by \(f\). For example, we may write \[\left.\frac{df}{dx}\right|_{x=2}=5.\]

Example 3.3. Find \(\frac{dy}{dx}\), if \(y=\frac{1}{x}.\)

Solution

Let \(f(x)=1/x\). Then \(f(x+h)=1/(x+h)\) and

\[\begin{aligned} \frac{dy}{dx} & =\lim_{h\to0}\frac{\frac{1}{x+h}-\frac{1}{x}}{h}\\ & =\lim_{h\to0}\frac{\frac{x-(x+h)}{x(x+h)}}{h}\tag{\small common denominator}\\ & =\lim_{h\to0}\frac{-\bcancel{h}}{\bcancel{h}x(x+h)}\tag{\small simplify}\\ & =\lim_{h\to0}\frac{-1}{x(x+h)}\\ & =-\frac{1}{x^{2}}\tag{\small substitute \ensuremath{h=0}}\end{aligned}\]



  1. Because the line that connects \(P\) and \(Q\) cuts the curve, it is called a “secant line.” The use of the word secant for this situation originates from the Latin secare, meaning to cut, and does not refer to the secant function in trigonometry.↩︎

  2. The equation of a line that passes through a given point \(P(a,b)\) with slope \(m\) is \(y-b=m\left(x-a\right).\)↩︎


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