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What is the definition of a derivative?
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The derivative of a function represents the rate at which the function's value changes as its input changes. It's the limit of the difference quotient as the interval approaches zero.
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How is the derivative of a function f(x) represented?
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The derivative of a function f(x) is commonly represented as f'(x) or df/dx.
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What is the Power Rule for differentiation?
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If f(x) = x^n, then f'(x) = nx^(n-1). This is known as the Power Rule.
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What is the derivative of a constant function?
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The derivative of a constant function is always zero, because constants do not change.
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How does the Chain Rule for differentiation work?
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The Chain Rule is used to differentiate compositions of functions. If h(x) = f(g(x)), then h'(x) = f'(g(x)) * g'(x).
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What is a critical point in calculus?
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A critical point of a function is where its derivative is zero or undefined, and it's often where a function changes from increasing to decreasing or vice versa.
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What does the derivative tell us about the graph of a function?
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The derivative provides information on the slope of the tangent line to the curve of the function at any given point.
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How can the derivative be used to find the local maximum and minimum of a function?
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By finding where the derivative is zero (critical points) and using the Second Derivative Test to determine if those points are maxima or minima.
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What is the Second Derivative Test?
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The Second Derivative Test is used to determine the concavity of a function and whether a critical point is a local maximum, minimum, or saddle point.
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How do you find the derivative of a product of two functions?
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Using the Product Rule: if u(x) and v(x) are functions, then the derivative of their product is u'(x)v(x) + u(x)v'(x).
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What is the Quotient Rule?
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The Quotient Rule is used to differentiate the quotient of two functions: if f(x) = g(x)/h(x), then f'(x) = (g'(x)h(x) - g(x)h'(x)) / [h(x)]^2.
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What is implicit differentiation?
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Implicit differentiation is used to find the derivative of functions that are not explicitly solved for one variable.
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Why is differentiation important in real-world applications?
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Differentiation is used to compute rates of change, such as velocity and acceleration, and to find optimal solutions in various fields like economics and engineering.
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What happens to the derivative of a function at an inflection point?
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At an inflection point, the second derivative changes sign, indicating a change in the direction of concavity of the function.
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Can all functions be differentiated?
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Not all functions can be differentiated. A function can be differentiated only if it is continuous and smooth (without sharp corners) at the point in question.
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