4.1.1 A Shortcut for Finding Derivatives

Question 1

Shortcut for Finding Derivatives

Answer 1

• Using the definition to find the derivative of a function is very time-consuming. However, when dealing with variables raised to rational powers, there is a shortcut you can use that makes finding derivatives easier. This shortcut is called the power rule.
• The power rule states that if N is a rational number, then the function f(x) = x^N is differentiable and f’(x) = Nx^N-1

Question 2

note

Answer 2

• This is a table of some functions and their derivatives.
• If you look carefully, you can see a pattern between the
  powers of the terms of the function and the powers of the terms of the derivative.
• You can also find a pattern between the powers of the terms of the functions and the constants of the terms of the derivatives.
• In each case, the power of the term of the derivative is one less than the power of the corresponding term of the function.
• Also, the constant multiple of each term of the derivative is equal to the constant multiple of the corresponding term of the function multiplied by the power of that term.
• This pattern gives rise to a shortcut called the power rule.
• The power rule works on any term made up of a variable raised to a rational power.
• To use the power rule, take the exponent of the original term and multiply it by the term. Then reduce the exponent by one.

Question 3

Suppose f(x)=x^−3. What is f′(x)?

Answer 3

f′(x)=−3x^-4

Question 4

You can use the power rule to take the derivative of functions with exponents expressed as:

Answer 4

• negative integers
• natural numbers
• negative fractions

Question 5

Suppose f(x)=x^5/2. What is the slope of the line tangent to f at x=4?

Answer 5

20

Question 6

The power rule is used to find the derivative of what sorts of functions?

Answer 6

Functions of x raised to a power.

Question 7

The power rule can be expressed as:

Answer 7

[xn]′=nx^n−1

Question 8

When using the power rule, the original coefficient:

Answer 8

Is multiplied by the original exponent

Question 9

Suppose f(x)=x^7/2.Find the equation of the line tangent to f(x)at (2,8√2).

Answer 9

y=(14√2)x−20√2

Question 10

Suppose f (x) = x^ −4/3. What is the slope of the line tangent to f at x = 2?

Answer 10

-^3√4 /6

Question 11

Suppose f (x) = −x ^−1. What is the slope of the line tangent to f at x = 3?

Answer 11

1/9

Question 12

Suppose f(x)=x7. What is the slope of the line tangent to f at x=2?

Answer 12

448

Question 13

Find the derivative of f if f (x) = x ^50.

Answer 13

f′(x)=50x^49

Question 14

When using the power rule, the original exponent:

Answer 14

is reduced by one