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Questions and Answers
What is the result when using the power rule on the function f(x) = x^5?
What is the result when using the power rule on the function f(x) = x^5?
The power rule can only be applied to positive integer powers.
The power rule can only be applied to positive integer powers.
False
What is the derivative of f(x) = x?
What is the derivative of f(x) = x?
1
To derive a function using the power rule, you bring down the ______ and then subtract ______ from it.
To derive a function using the power rule, you bring down the ______ and then subtract ______ from it.
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What does the product rule state regarding derivatives?
What does the product rule state regarding derivatives?
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The quotient rule can be derived directly from the product rule.
The quotient rule can be derived directly from the product rule.
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What are the three components to apply the product rule for three functions?
What are the three components to apply the product rule for three functions?
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The product rule states that if f(x) = u(x) * v(x), then ______ = u'(x) * v(x) + u(x) * v'(x).
The product rule states that if f(x) = u(x) * v(x), then ______ = u'(x) * v(x) + u(x) * v'(x).
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Match the following terms with their descriptions:
Match the following terms with their descriptions:
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What common mistake is made when deriving the derivative without using the product rule?
What common mistake is made when deriving the derivative without using the product rule?
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Both the product rule and the quotient rule are derived from the definition of a derivative.
Both the product rule and the quotient rule are derived from the definition of a derivative.
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Define the derivative using its foundational limit concept.
Define the derivative using its foundational limit concept.
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To find the derivative of a function that is the ratio of ______ functions, apply the quotient rule.
To find the derivative of a function that is the ratio of ______ functions, apply the quotient rule.
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In the product rule, what term approaches as h approaches 0?
In the product rule, what term approaches as h approaches 0?
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What is the result of applying the power rule to the function f(x) = x^3?
What is the result of applying the power rule to the function f(x) = x^3?
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The quotient rule is used to find derivatives of products of functions.
The quotient rule is used to find derivatives of products of functions.
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What does the derivative of the function f(x) = x^0 equal?
What does the derivative of the function f(x) = x^0 equal?
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To apply the power rule, we bring down the _____ and then subtract _____ from it.
To apply the power rule, we bring down the _____ and then subtract _____ from it.
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Which rule can be used to find the derivative of the function f(x) = x^(-2)?
Which rule can be used to find the derivative of the function f(x) = x^(-2)?
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Briefly explain how the power rule is derived.
Briefly explain how the power rule is derived.
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Match the following functions with their derivatives using the power rule:
Match the following functions with their derivatives using the power rule:
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The slope of the line represented by the function f(x) = x is _____ .
The slope of the line represented by the function f(x) = x is _____ .
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Study Notes
Derivatives of Functions: Power, Product, and Quotient Rules
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Power Rule: Finds the derivative of a variable raised to a power. Bring the exponent down and times it by the variable, then reduce the exponent by 1. Example: f(x) = x²; f'(x) = 2x.
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Proof of Power Rule: Deriving from the derivative definition:
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Substitute f(x) into the derivative definition.
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Simplify the expression for the limit as h approaches 0.
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Result matches the power rule.
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For f(x)=x, power rule gives f'(x)=1. A graph of f(x)=x is a line with slope 1, hence derivative is 1.
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Power Rule Applicability: Applies to positive, negative, and fractional exponents including radical functions (express exponents as fractions initially).
Product Rule
- Problem. Finding derivatives of the product of two functions.
- Method: The derivative of the product of functions is not the product of their derivatives. The product rule solves this issue: (u(x)v(x))' = u'(x)v(x) + u(x)v'(x).
Proof of Product Rule
- Method 1 (Direct): Start with the derivative definition, f'(x) = lim(h->0) [f(x + h) - f(x)]/h. Substitute f(x) = u(x)v(x), and manipulate the resulting expression to separate the limit terms and arrive at the rule.
Quotient Rule
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Problem: Finding the derivative of a function that is the ratio of two functions.
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Method: (u(x)/v(x))' = [u'(x)v(x) - u(x)v'(x)] / [v(x)]²
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Derivation Shortcut: Using the product rule on u(x)/v(x) to derive the quotient rule.
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Example: Tangent function derivative (tan(x)) can be found using the quotient rule and known derivatives of sin(x) and cos(x).
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Description
This quiz covers the fundamental rules for finding derivatives in calculus, specifically focusing on the Power and Product Rules. Understand how to apply these rules through definitions, examples, and proofs, enhancing your skills in differentiation.
