3B Survey Review

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What is the chain rule in calculus?

Back

The chain rule is a formula for computing the derivative of the composition of two or more functions. It states that if you have a function y = f(g(x)), then the derivative y' = f'(g(x)) * g'(x).

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How do you find the derivative of a power function?

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To find the derivative of a power function f(x) = x^n, use the power rule: f'(x) = n*x^(n-1).

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What is the derivative of e^(kx)?

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The derivative of e^(kx) with respect to x is k*e^(kx), where k is a constant.

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How do you find the equation of a tangent line at a point on a curve?

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To find the equation of a tangent line at a point (a, f(a)), first find the derivative f'(a) to get the slope. Then use the point-slope form: y - f(a) = f'(a)(x - a).

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What is implicit differentiation?

Back

Implicit differentiation is a technique used to find the derivative of a function defined implicitly by an equation involving both x and y, without solving for y explicitly.

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What is the product rule in calculus?

Back

The product rule is used to find the derivative of the product of two functions. If u(x) and v(x) are functions, then (uv)' = u'v + uv'.

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What is the quotient rule in calculus?

Back

The quotient rule is used to find the derivative of a quotient of two functions. If u(x) and v(x) are functions, then (u/v)' = (u'v - uv')/v^2.

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What is the significance of the first derivative test?

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The first derivative test is used to determine the local maxima and minima of a function by analyzing the sign of the derivative before and after critical points.

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What is a critical point in calculus?

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A critical point occurs where the derivative of a function is zero or undefined, indicating potential local maxima, minima, or points of inflection.

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How do you apply the second derivative test?

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The second derivative test helps determine the concavity of a function at critical points. If f''(x) > 0, the function is concave up (local minimum); if f''(x) < 0, it is concave down (local maximum).