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What is the definition of a derivative?
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The derivative of a function is a measure of how the function value changes as its input changes. It represents an instantaneous rate of change and is the slope of the tangent line to the function at a point.
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How is the derivative of a function denoted?
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The derivative of a function f(x) can be denoted as f'(x), df/dx, or Df(x).
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What is the power rule for differentiation?
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The power rule states that d/dx [x^n] = nx^(n-1) for any real number n.
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What is the derivative of a constant function?
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The derivative of a constant function is 0, as constant functions do not change.
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How do you differentiate a product of two functions?
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The derivative of a product of two functions, u(x) and v(x), is given by the product rule: (uv)' = u'v + uv'.
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What is the chain rule in calculus?
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The chain rule states that if a variable z depends on y, which depends on x, then the derivative of z with respect to x is dz/dx = (dz/dy) * (dy/dx).
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How is the derivative of a sum of functions determined?
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The derivative of a sum of functions is the sum of their derivatives: d/dx [u(x) + v(x)] = u'(x) + v'(x).
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What does the derivative tell you about the graph of a function?
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The derivative provides information on the slope or steepness of the graph at any given point, indicating whether the function is increasing or decreasing.
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What is the geometric interpretation of a derivative at a point?
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The derivative at a point represents the slope of the tangent line to the curve of the function at that point.
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How do you differentiate a quotient of two functions?
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The derivative of a quotient u(x)/v(x) is given by the quotient rule: (u/v)' = (u'v - uv')/v^2.
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What is implicit differentiation?
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Implicit differentiation is used to find the derivative of a function expressed in terms of two or more variables, without explicitly solving for one variable in terms of the others.
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When do you use the concept of higher-order derivatives?
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Higher-order derivatives are used when examining the rate of change of the rate of change, often for studying concavity and the behavior of functions beyond just their slope.
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What is the meaning of the second derivative?
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The second derivative measures the rate of change of the first derivative, providing information on the concavity of the original function.
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How is the second derivative used in determining concavity?
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If the second derivative of a function is positive, the function is concave up. If it is negative, the function is concave down.
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What role does the derivative play in finding local extrema?
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The derivative helps find local maxima and minima by identifying critical points where the first derivative is zero or undefined, and then using the second derivative test to determine the nature of these points.
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