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Questions and Answers
What is a primary purpose of derivatives in mathematics?
What is a primary purpose of derivatives in mathematics?
Which of the following represents the derivative of a function?
Which of the following represents the derivative of a function?
If a function has a derivative of zero at a point, what can be inferred?
If a function has a derivative of zero at a point, what can be inferred?
Which method is commonly used to find derivatives?
Which method is commonly used to find derivatives?
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What does the notation f'(x) typically represent?
What does the notation f'(x) typically represent?
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Study Notes
Derivatives
- Finding the tangent: Derivatives are used to find the tangent line to a function's graph.
- Secant lines: The slope of a secant line approximating the tangent line is calculated as [f(a + h) - f(a)] / h.
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Tangent line slope: The slope of the tangent line is calculated as the limit of the secant line slope as h approaches 0.
- Formula: lim (h→0) [f(a + h) - f(a)] / h
Definition of the Derivative
- General point: The slope of a tangent at any general point 'x' is given as the limit.
- Formula: m = lim (h→0) [f(x + h) - f(x)] / h
Definition: The Derivative of a Function at a Point
- Formula: f'(x) = lim (h→0) [f(x + h) - f(x)] / h
Differentiation Rules
- Constant Rule: d/dx (c) = 0
- Constant Multiple Rule: d/dx (cf(x)) = c * f'(x)
- Power Rule: f(x) = xn => f'(x) = nxn-1 (where n is any real number)
- Sum/Difference Rule: d/dx (f(x) ± g(x)) = f'(x) ± g'(x)
- Product Rule: d/dx (f(x) * g(x)) = f'(x)g(x) + f(x)g'(x)
- Quotient Rule: d/dx (f(x)/g(x)) = [f'(x)g(x) - f(x)g'(x)] / g2(x) (provided g(x) ≠ 0)
The Chain Rule
- If y = f(u) and u = g(x), then dy/dx = dy/du * du/dx
- [f(g(x))]' = f'(g(x)) * g'(x)
Logarithmic Differentiation
- Takes the natural logarithm of both sides of the equation to simplify complex functions.
- Uses properties of logarithms to make the derivative easier.
Implicit Differentiation
- Used when the relationship between x and y is not in the form y = f(x).
- When differentiating with respect to 'x', recall that dy/dx is commonly written as 'y'.
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Description
Test your understanding of derivatives in calculus, including how to find tangent lines, secant lines, and apply differentiation rules. This quiz covers key concepts and formulas related to the definition of the derivative and differentiation techniques.
