Product and Quotient Rules (Derivatives)

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What is the Product Rule for derivatives?

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The Product Rule states that if you have two functions u(x) and v(x), the derivative of their product is given by: \(\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)\).

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What is the Quotient Rule for derivatives?

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The Quotient Rule states that if you have a function that is the quotient of two functions u(x) and v(x), the derivative is given by: \(\frac{d}{dx}\left[\frac{u(x)}{v(x)}\right] = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}\).

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How do you find the slope of the tangent line to a curve at a given point?

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To find the slope of the tangent line at a point (a, f(a)), compute the derivative f'(x) and evaluate it at x = a.

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What is the significance of the derivative in calculus?

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The derivative represents the rate of change of a function with respect to its variable, indicating how the function behaves locally.

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What is a tangent line?

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A tangent line to a curve at a given point is a straight line that touches the curve at that point and has the same slope as the curve at that point.

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What is the chain rule in differentiation?

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The Chain Rule states that if a function y = f(g(x)) is composed of two functions, then its derivative is: \(\frac{dy}{dx} = f'(g(x))g'(x)\).