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Derivatives

Definition 3.1.1.  The derivative of a function \( f \) at a number \( a \), denoted by \( f'(a) \), is

\( \displaystyle f'(a) = \lim_{h \to 0}\frac{f(a + h) - f(a)}{h} \)

If \( x = a + h \), then

\( \displaystyle f'(a) = \lim_{x \to a}\frac{f(x) - f(a)}{x - a} \)

Definition 3.1.2.  If we replace \( a \) by a variable \( x \) in Definition 3.1.1, we obtain

\( \displaystyle f'(x) = \lim_{h \to 0}\frac{f(x + h) - f(x)}{h} \)

Given any number \( x \) for which this limit exists, we assign to \( x \) the number \( f'(x) \). So we can regard \( f' \) as a new function, called the derivative of \( f \).

Remark 3.1.3.  If we use the traditional notation \( y = f(x) \) to indicate that the independent variable is \( x \) and the dependent variable is \( y \), then some common alternative notations for the derivative are as follows:

\( \displaystyle f'(x) = y' = {\operatorname{d}\!y\over\operatorname{d}\!x} = {\operatorname{d}\!f\over\operatorname{d}\!x} = {\operatorname{d}\over\operatorname{d}\!x}f(x) = Df(x) = D_{x}f(x) \)

The symbols \( D \) and \( \displaystyle {\operatorname{d}\over\operatorname{d}\!x} \) are called differentiation operators because they indicate the operation of differentiation, which is the process of calculating a derivative. The symbol \( \displaystyle {\operatorname{d}\!y\over\operatorname{d}\!x} \), which was introduced by Leibniz, should not be regarded as a ratio (for the time being); it is simply a synonym for \( f'(x) \).

If we want to indicate the value of a derivative \( \displaystyle {\operatorname{d}\!y\over\operatorname{d}\!x} \) at a specific number \( a \), we use the notation

\( \displaystyle {\operatorname{d}\!y\over\operatorname{d}\!x} \Bigg\vert_{\ x = a} \qquad \text{or} \qquad {\operatorname{d}\!y\over\operatorname{d}\!x} \Bigg]_{x = a} \)

which is a synonym for \( f'(a) \).

Example 3.1.4.  If \( f(x) = x^{3} - x \), find a formula for \( f'(x) \).

Solution.

\( \displaystyle \begin{align} f'(x) &= \lim_{h \to 0}\frac{f(x + h) - f(x)}{h} = \lim_{h \to 0}\frac{[(x + h)^{3} - (x + h)] - [x^{3} - x]}{h} \\[7pt] &= \lim_{h \to 0}\frac{x^{3} + 3x^{2}h + 3xh^{2} + h^{3} - x - h - x^{3} + x}{h} \\[7pt] &= \lim_{h \to 0}\frac{3x^{2}h + 3xh^{2} + h^{3} - h}{h} \\[7pt] &= \lim_{h \to 0}(3x^{2} + 3xh + h^{2} - 1) = 3x^{2} - 1 \end{align} \)

\( \Box \)

Theorem 3.1.5.  The tangent line to the curve \( y = f(x) \) at the point \( P(a, f(a)) \) is the line through \( P \) with slope

\( \displaystyle m = \lim_{x \to a} \frac{f(x) - f(a)}{x - a} \)

provided that this limit exists. And if \( x = a + h \), the slope of the tangent line is

\( \displaystyle m = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h} \)

Figure 1

Remark 3.1.6.  We can now say that the tangent line to \( y = f(x) \) at \( (a, f(a)) \) is the line through \( (a, f(a)) \) whose slope is equal to \( f'(a) \), the derivative of \( f \) at \( a \). In other words,

\( \displaystyle m = \lim_{x \to a} \frac{f(x) - f(a)}{x - a} = f'(a) \)

Therefore, if we use the point-slope form of the equation of a line, we can write an equation of the tangent line to the curve \( y = f(x) \) at the point \( (a, f(a)) \):

\( \displaystyle y - f(a) = f'(a)(x - a) \)

Definition 3.1.7.  A function \( f \) is differentiable at \( a \) if \( \displaystyle f'(a) = \lim_{x \to a}\frac{f(x) - f(a)}{x - a} \) exists. It is differentiable on an open interval \( (a,\ b) \) if it is differentiable at every number in the interval.

Theorem 3.1.8.  If \( f \) is differentiable at \( a \), then \( f \) is continuous at \( a \).

Proof.  To prove that \( f \) is continuous at \( a \), we have to show that \( \displaystyle \lim_{x \to a}f(x) = f(a) \). We do this by showing that the difference \( f(x) - f(a) \) approaches \( 0 \).

The given information is that \( f \) is differentiable at \( a \), that is,

\( \displaystyle f'(a) = \lim_{x \to a}\frac{f(x) - f(a)}{x - a} \)

exists. To connect the given and the unknown, we divide and multiply \( f(x) - f(a) \) by \( x - a \) (which we can do when \( x \neq a \)) :

\( \displaystyle f(x) - f(a) = \frac{f(x) - f(a)}{x - a}(x - a) \)

Thus, we can write

\( \displaystyle \begin{align} \lim_{x \to a}\left[ f(x) - f(a) \right] &= \lim_{x \to a}\frac{f(x) - f(a)}{x - a}(x - a) \\[7pt] &= \lim_{x \to a}\frac{f(x) - f(a)}{x - a} \cdot \lim_{x \to a}(x - a) \\[7pt] &= f'(a) \cdot 0 = 0 \end{align} \)

Therefore,

\( \displaystyle \begin{align} \lim_{x \to a}f(x) &= \lim_{x \to a}\left[ f(a) + f(x) - f(a) \right] \\[7pt] &= \lim_{x \to a}f(a) + \lim_{x \to a}\left[ f(x) - f(a) \right] \\[7pt] &= f(a) + 0 = f(a) \end{align} \)

Thus, \( f \) is continuous at \( a \).

\( \Box \)

Definition 3.1.9.  The second derivative of \( y = f(x) \) is

\( \displaystyle f''(x) = {\operatorname{d}\over\operatorname{d}\!x} \left( {\operatorname{d}\!y\over\operatorname{d}\!x} \right) = {\operatorname{d}^{2}\!y\over\operatorname{d}\!x^{2}} \)

The third derivative of \( y = f(x) \) is

\( \displaystyle f'''(x) = {\operatorname{d}\over\operatorname{d}\!x} \left( {\operatorname{d}^{2}\!y\over\operatorname{d}\!x^{2}} \right) = {\operatorname{d}^{3}\!y\over\operatorname{d}\!x^{3}} \)

\( \begin{matrix} \vdots \\[7pt] \end{matrix} \)

The \( n \)th derivative of \( y = f(x) \) is

\( \displaystyle f^{(n)}(x) = {\operatorname{d}^{n}\!y\over\operatorname{d}\!x^{n}} \)

Rates of Change

Definition 3.2.1.  Suppose \( y \) is a quantity that depends on another quantity \( x \). Thus \( y \) is a function of \( x \) and we write \( y = f(x) \). If \( x \) changes from \( x_1 \) to \( x_2 \), then the change in \( x \) (also called the increment of \( x \)) is

\( \Delta x = x_{2} - x_{1} \)

and the corresponding change in \( y \) is

\( \Delta y = f(x_{2}) - f(x_{1}) \)

Definition 3.2.2.  The difference quotient

\( \displaystyle \frac{\Delta y}{\Delta x} = \frac{f(x_{2}) - f(x_{1})}{x_{2} - x_{1}} \)

is called the average rate of change of \( y \) with respect to \( x \) over the interval \( \left[ x_{1},\ x_{2} \right] \). And the limit of the average rate of change

\( \displaystyle \lim_{\Delta x \to 0}\! \frac{\Delta y}{\Delta x} = \lim_{x_{2} \to x_{1}}\! \frac{f(x_{2}) - f(x_{1})}{x_{2} - x_{1}} \)

is called the (instantaneous) rate of change of \( y \) with respect to \( x \) at \( x = x_{1} \).

Definition 3.2.3.  If \( s = f(t) \) is the position function of a moving object, the average velocity of the object from \( t = t_{0} \) to \( t = t_{0} + \Delta t \) is

\( \displaystyle v_{\text{av}} = \frac{s(t_{0} + \Delta t) - s(t_{0})}{\Delta t} \)

and the velocity (or instantaneous velocity) at time \( t = t_{0} \) is

\( \displaystyle v(t_{0}) = {\operatorname{d}\!s\over\operatorname{d}\!t} \Bigg\vert_{\ t = t_{0}} = \lim_{\Delta t \to 0}\!\frac{s(t_{0} + \Delta t) - s(t_{0})}{\Delta t} \)

and the acceleration at time \( t = t_{0} \) is

\( \displaystyle a(t_{0}) = {\operatorname{d}^{2}\!s\over\operatorname{d}\!t^{2}} \Bigg\vert_{\ t = t_{0}} = \lim_{\Delta t \to 0}\!\frac{v(t_{0} + \Delta t) - v(t_{0})}{\Delta t} \)

and the jerk at time \( t = t_{0} \) is

\( \displaystyle j(t_{0}) = {\operatorname{d}^{3}\!s\over\operatorname{d}\!t^{3}} \Bigg\vert_{\ t = t_{0}} = \lim_{\Delta t \to 0}\!\frac{a(t_{0} + \Delta t) - a(t_{0})}{\Delta t}\)

Example 3.2.4.  The position of a particle is given by the equation

\( \displaystyle s = f(t) = t^{3} - 6t^{2} + 9t \)

where \( t \) is measured in seconds and \( s \) in meters. Find the velocity at time \( t \).

Solution.  The velocity function is the derivative of the position function.

\( \displaystyle v(t) = {\operatorname{d}\!s\over\operatorname{d}\!t} = 3t^{2} - 12t + 9 \)

\( \Box \)

The body text and the figures were excerpted from James Stewart, Calculus 8E.