AP Calculus AB and BC: Chapter 2 – Differentiation :2.1 -Definition of Derivatives and the Power Rule Study Notes

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

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