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Questions and Answers
What is the purpose of related rates in calculus?
What is the purpose of related rates in calculus?
How can derivatives assist in curve sketching?
How can derivatives assist in curve sketching?
What is the goal of optimization using derivatives?
What is the goal of optimization using derivatives?
What does Newton's Method utilize to approximate solutions?
What does Newton's Method utilize to approximate solutions?
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How are derivatives used in economics?
How are derivatives used in economics?
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What is the derivative of the constant function f(x) = 5?
What is the derivative of the constant function f(x) = 5?
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Using the power rule, what is the derivative of f(x) = x^4?
Using the power rule, what is the derivative of f(x) = x^4?
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What is the derivative of the function f(x) = 2x^3 + 3x - 5?
What is the derivative of the function f(x) = 2x^3 + 3x - 5?
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What is the second derivative of the function f(x) = x^3?
What is the second derivative of the function f(x) = x^3?
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If f(x) = cos(x), what is f'(x)?
If f(x) = cos(x), what is f'(x)?
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What is the derivative of the function f(x) = e^(3x)?
What is the derivative of the function f(x) = e^(3x)?
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What is the derivative of f(x) = sin(x) + ln(x)?
What is the derivative of f(x) = sin(x) + ln(x)?
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What is the derivative of the piecewise function defined as f(x) = {x^2 for x < 0, x + 1 for x ≥ 0}?
What is the derivative of the piecewise function defined as f(x) = {x^2 for x < 0, x + 1 for x ≥ 0}?
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Study Notes
Basic Differentiation Rules
- Constant Rule: The derivative of a constant is zero. If f(x) = c, then f'(x) = 0.
- Power Rule: The derivative of xn is nxn-1. This applies to any constant power, positive, negative, or fractional.
- Constant Multiple Rule: The derivative of a constant times a function is the constant times the derivative of the function. If f(x) = cg(x), then f'(x) = cg'(x).
- Sum/Difference Rule: The derivative of the sum (or difference) of two functions is the sum (or difference) of their derivatives. If f(x) = g(x) ± h(x), then f'(x) = g'(x) ± h'(x).
- Product Rule: The derivative of the product of two functions is the first function times the derivative of the second, plus the second function times the derivative of the first. If f(x) = g(x)*h(x), then f'(x) = g(x)h'(x) + h(x)g'(x).
- Quotient Rule: The derivative of the quotient of two functions is the denominator times the derivative of the numerator, minus the numerator times the derivative of the denominator, all over the denominator squared. If f(x) = g(x)/h(x), then f'(x) = [h(x)g'(x) - g(x)h'(x)] / [h(x)]2.
Trigonometric Functions
- sin(x): cos(x)
- cos(x): -sin(x)
- tan(x): sec2(x)
- csc(x): -csc(x)cot(x)
- sec(x): sec(x)tan(x)
- cot(x): -csc2(x)
Chain Rule
- The chain rule is used to differentiate composite functions. If f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x). Differentiate the outer function, leaving the inner function alone, then multiply by the derivative of the inner function.
Implicit Differentiation
- Used to find the derivative of a function where y is defined implicitly as a function of x in an equation.
- Treat y as a function of x and apply the chain rule whenever differentiating y with respect to x.
- Isolate dy/dx.
Exponential and Logarithmic Functions
- ex: ex
- ax: axln(a)
- ln(x): 1/x
- loga(x): (1/x) / ln(a)
Higher Order Derivatives
- The second derivative of a function, f''(x), is the derivative of the first derivative, f'(x).
- Higher order derivatives are found by repeatedly differentiating the function.
Derivatives of Inverse Trigonometric Functions
- arcsin(x): 1/√(1-x2)
- arccos(x): -1/√(1-x2)
- arctan(x): 1/(1+x2)
Derivatives of a piecewise function
- A piecewise function is defined by different expressions in different subintervals.
- Evaluate the derivative separately for each interval.
Derivatives applied to problems in applications
- Related Rates: Use derivatives to find how the rate of change of one quantity is related to the rate of change of another quantity. This often involves situations with multiple changing quantities.
- Optimization: Use derivatives to find maximum or minimum values of functions.
- Curve Sketching: Derivatives provide information about the slope of a function, analyzing monotonicity (increasing/decreasing) and concavity (upward/downward curvature). This leads to critical points (max/min) and inflection points.
- Newton's Method: Use derivatives to approximate solutions to equations.
- Marginals in Economy: Use derivatives to find marginal cost, marginal revenue, and marginal profit.
Studying That Suits You
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Description
Test your understanding of the fundamental rules of differentiation. This quiz covers essential concepts such as the Constant Rule, Power Rule, and Product Rule. Perfect for students looking to reinforce their calculus knowledge.
