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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?
- To determine rates of change. (correct)
- To factor polynomials.
- To simplify fractions.
- To calculate integral values.
Which of the following represents the derivative of a function?
Which of the following represents the derivative of a function?
- The maximum point of the function.
- The average value of the function.
- The slope of the tangent line. (correct)
- The area under the curve.
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?
- The function is constant at that point.
- The function has a maximum or minimum at that point. (correct)
- The function is increasing at that point.
- The function is decreasing at that point.
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.
- 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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