Flashcards in 4.1.1 A Shortcut for Finding Derivatives Deck (14):
Shortcut for Finding Derivatives
• 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
- 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.
Suppose f(x)=x^−3. What is f′(x)?
You can use the power rule to take the derivative of functions with exponents expressed as:
- negative integers
- natural numbers
- negative fractions
Suppose f(x)=x^5/2. What is the slope of the line tangent to f at x=4?
The power rule is used to find the derivative of what sorts of functions?
Functions of x raised to a power.
The power rule can be expressed as:
When using the power rule, the original coefficient:
Is multiplied by the original exponent
Suppose f(x)=x^7/2.Find the equation of the line tangent to f(x)at (2,8√2).
Suppose f (x) = x^ −4/3. What is the slope of the line tangent to f at x = 2?
Suppose f (x) = −x ^−1. What is the slope of the line tangent to f at x = 3?
Suppose f(x)=x7. What is the slope of the line tangent to f at x=2?
Find the derivative of f if f (x) = x ^50.