Definition of Derivatives
The derivative of a function f at x , denoted by f′(x) is
\({f}’\left ( x \right )=\underset{h\rightarrow 0}{lim}\frac{f\left ( x+h \right )-f\left ( x \right )}{h}\)
if the limit exists.
If we replace x = a in the above equation, the derivative of a function f at a number a , denoted by f′(a) , is
\({f}’\left ( x \right )=\underset{h\rightarrow 0}{lim}\frac{f\left ( a+h \right )-f\left ( a \right )}{h}\).
If we write x = a + h, then h = x – a and h approaches 0 if and only if x approaches a . Therefore, an equivalent way of stating the definition of derivative is
\({f}’\left ( x \right )=\underset{x\rightarrow a}{lim}\frac{f\left ( x\right )-f\left ( a \right )}{x-a}\).
The process of finding the derivative of a function is called differentiation. In addition to f′(x), which is read as “f prime of x ,” other notations such as \(\frac{dy}{dx}, {y}’,\) and \(\frac{d}{dx}\left [ f\left ( x \right ) \right ]\) are used to denote the derivative of y = f (x) .
Differentiability Implies Continuity
A function f is differentiable at a only if f′(a) exists. If a function is differentiable at x = a, then f is continuous at x = a. Continuity, however, does not imply differentiability.
The derivative f′(a) is the instantaneous rate of change of f(x) with respect to x when x = a.
One-Sided Derivatives
The left-hand derivative of f at a is defined by \({f}’\left ( a^{^{-}} \right )=\underset{h\rightarrow 0^{-}}{lim}\frac{f\left ( a+h \right )-f\left ( a \right )}{h}\), if the limit exist.
The right-hand derivative of f at a is defined by \({f}’\left ( a^{^{+}} \right )=\underset{h\rightarrow 0^{+}}{lim}\frac{f\left ( a+h \right )-f\left ( a \right )}{h}\), if the limit exist.
f′(a) exists if and only if these one sided derivatives exist and are equal.
There are three possible ways for f not to be differentiable at x = a.
Basic Differentiation Rules
The Constant Rule
The derivative of a constant function is 0. \(\frac{d}{dx}\left [ c \right ]=0\)
The Power Rule
If n is any real number, then \(\frac{d}{dx}\left [ x^{^{n}}\right ]=nx^{n-1}.\)
Example 1
- What is \(\underset{h\rightarrow 0}{lim}\frac{\left ( 3+h \right )^{4}-81}{h}?\)
▶️Answer/Explanation
Solution
\(\underset{h\rightarrow 0}{lim}\frac{\left ( 3+h \right )^{4}-81}{h}=\underset{h\rightarrow 0}{lim}\frac{\left ( 3+h \right )^{4}-3^{4}}{h}={f}’\left ( 3 \right )\), where \(f\left ( x \right )=x^{4}\) and a = 3.
\({f}’\left ( x \right )=4x^{3}\) Power Rule
\({f}’\left ( 3 \right )=4\left (3 \right )^{3}=108\)
Example 2
- Find the derivative of \(f\left ( x \right )=x^{3}-2x+\frac{1}{x}+5\) at x=2.
▶️Answer/Explanation
Solution
\(f\left ( x \right )=x^{3}-2x+x^{-1}+5\) \(1/x=x^{-1}\)
\({f}’\left ( x \right )=3x^{3-1}-2x^{1-1}+\left ( -1 \right )x^{-1-1}+0\) The power rule and the constant rule
=3x^{2}-2x^{0}-x^{-2}=3x^{2}-2-\frac{1}{x^{2}}
\({f}’\left ( 2 \right )=3\left ( 2 \right )^{2}-2\frac{1}{\left ( 2 \right )^{2}}=\frac{39}{4}\)
Example 3
- Find a function f and a number a such that
\({f}’\left ( a \right )=\underset{h\rightarrow 0}{lim}\frac{sin\left ( \frac{\pi }{6}+h \right )-\frac{1}{2}}{h}.\)
▶️Answer/Explanation
Solution
\({f}’\left ( a \right )=\underset{h\rightarrow 0}{lim}\frac{sin\left ( \frac{\pi }{6}+h \right )-\frac{1}{2}}{h}=\underset{h\rightarrow 0}{lim}\frac{f\left ( a+h \right )-f\left ( a \right )}{h}\)
From the above equation we can conclude that
\(f\left ( a+h \right )=sin\left ( \frac{\pi }{6}+h \right )\) and \(f\left ( a \right )=\frac{1}{2}=sin\left ( \frac{\pi }{6} \right ).\)
So \(a=\frac{\pi }{6}\) and \(f\left ( x \right )sinx.\)
Example 4
- The graph of f is shown in the figure below. For what values of x , −1 < x < 3, is f not differentiable?
▶️Answer/Explanation
Solution
f is not differentiable at x = 0 since the graph is discontinuous at x = 0 .
f is not differentiable at x = 1 since the graph has a corner, where the one-sided derivatives differ. (At x = 1 , f′ (1–) > 0 and f′ (1+) < 0 .)
Exercises – Definition of Derivatives and the Power Rule
Multiple Choice Questions
- \(\underset{h\rightarrow 0}{lim}\frac{\sqrt[3]{8+h-2}}{h}=\)
(A) \(\frac{1}{12}\) (B) \(\frac{1}{4}\) (C) \(\frac{\sqrt[3]{2}}{2}\) (D) \(\sqrt[3]{2}\) (E) 2
▶️Answer/Explanation
Ans:
1. A
- 2. \(\underset{h\rightarrow 0}{lim}\frac{\left ( 2+h \right )^{5}-32}{h}\) is
(A) f ′(5) , where \(f\left ( x \right )=x^{2}\)
(B) f ′(2) , where \(f\left ( x \right )=x^{5}\)
(C) f ′(5) , where \(f\left ( x \right )=2^{x}\)
(D) f ′(2) , where \(f\left ( x \right )=2^{x}\)
▶️Answer/Explanation
Ans:
2. B
\(f\left ( x \right )=\left\{\begin{matrix}1-2x, & if x \leq 1\\-x^{2}, & if x> 1\end{matrix}\right.\)
3. Let f be the function given above. Which of the following must be true?
I. \(\underset{x\rightarrow 1}{lim}f\left ( x \right )\) exists.
II. f is continuous at x = 1 .
III. f is differentiable at x = 1 .
(A) I only
(B) I and II only
(C) II and III only
(D) I, II, and III
▶️Answer/Explanation
Ans:
3. D
4. What is the instantaneous rate of change at x = −1 of the function \(f\left ( x \right )=-\sqrt[3]{x^{2}}?\)
(A) \(-\frac{2}{3}\) (B) \(-\frac{1}{3}\) (C) \(\frac{1}{3}\) (D) \(\frac{2}{3}\)
▶️Answer/Explanation
Ans:
4. D
5. The graph of a function f is shown in the figure above. Which of the following statements must be false?
(A) f (x) is defined for 0 ≤ x ≤ b .
(B) f (b) exists.
(C) f′(b) exists.
(D) \(\underset{x\rightarrow a}{lim}\) f′(x) exists.
▶️Answer/Explanation
Ans:
5. C
6. If f is a differentiable function, then f ′(1) is given by which of the following?
I. \(\underset{h\rightarrow 0}{lim}\frac{f\left ( 1+h \right )-f\left ( 1 \right )}{h}\)
II. \(\underset{x\rightarrow 1}{lim}\frac{f\left ( x \right )-f\left ( 1 \right )}{x-1}\)
III. \(\underset{x\rightarrow 0}{lim}\frac{f\left ( x+h \right )-f\left ( x \right )}{h}\)
(A) I only (B) II only (C) I and II only (D) I and III only
▶️Answer/Explanation
Ans:
6. C
7. The graph of a function f is shown in the figure above. At how many points in the interval a < x < j is f ′ not defined?
(A) 3 (B) 4 (C) 5 (D) 6
▶️Answer/Explanation
Ans:
7. B
Free Response Questions
8. Let f be the function defined by \(f\left ( x \right )=\left\{\begin{matrix}mx^{2}-2 & if x\leq 1\\k\sqrt{x} & if x> 1\end{matrix}\right.\) . If f is differentiable at x = 1 , what are the values of k and m ?
▶️Answer/Explanation
Ans:
8. m = −2/3 , k = −8/3
9. Let f be a function that is differentiable throughout its domain and that has the following properties.
(1) \(f\left ( x+y \right )=f\left ( x \right )+x^{3}y-xy^{3}-f\left ( y \right )\)
(2) \(\underset{x\rightarrow 0}{lim}\frac{f\left ( x \right )}{x}=1\)
Use the definition of the derivative to show that \({f}’\left ( x \right )=x^{3}-1.\)
▶️Answer/Explanation
Ans:
9.
10. Let f be the function defined by
\(f\left ( x \right )=\left\{\begin{matrix}x+2 & for x\leq 0\\\frac{1}{2}\left ( x+2 \right )^{2} & for x> 0.\end{matrix}\right.\)
(a) Find the left-hand derivative of f at x = 0 .
(b) Find the right-hand derivative of f at x = 0 .
(c) Is the function f differentiable at x = 0 ? Explain why or why not.
(d) Suppose the function g is defined by
\(g\left ( x \right )=\left\{\begin{matrix}x+2 & for x\leq 0\\a\left ( x+b \right )^{2} & for x> 0,\end{matrix}\right.\)
where a and b are constants. If g is differentiable at x = 0 , what are the values of a and b ?
▶️Answer/Explanation
Ans:
10. (a) 1
10. (b) 2
10. (c) No
10. (d) a = 1/8 , b = 4