Application of Derivatives

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dy/dx represents the rate of change of quantity y with respect to quantity x. If y = f(x), then dy/dx (or f'(x)) gives how fast y changes when x changes. At a specific point x₀, dy/dx|x=x₀ represents …

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Using the chain rule: dy/dx = (dy/dt)/(dx/dt), provided dx/dt ≠ 0. This formula helps find the rate of change of y with respect to x when both variables are functions of a third variable t.

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A function f is increasing on interval I if for any two points x₁ < x₂ in I, we have f(x₁) < f(x₂). In other words, as x increases, f(x) also increases. Mathematically: x₁ < x₂ ⟹ f(x₁) < f(x₂) for all…

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A function f is decreasing on interval I if for any two points x₁ < x₂ in I, we have f(x₁) > f(x₂). In other words, as x increases, f(x) decreases. Mathematically: x₁ < x₂ ⟹ f(x₁) > f(x₂) for all x₁, …

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For a differentiable function f on interval I: • If f'(x) > 0 for all x ∈ I, then f is increasing on I • If f'(x) < 0 for all x ∈ I, then f is decreasing on I • If f'(x) = 0 for all x ∈ I, then f is c…

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A point c in the domain of function f is called a critical point if either: 1. f'(c) = 0, or 2. f is not differentiable at c Critical points are important because local maxima and minima can only occu…

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Local Maximum: Point c is a local maximum if there exists h > 0 such that f(c) ≥ f(x) for all x in (c-h, c+h), x ≠ c. Local Minimum: Point c is a local minimum if there exists h > 0 such that f(c) ≤ …

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For a continuous function f at critical point c: • If f'(x) changes from positive to negative as x passes through c, then c is a local maximum • If f'(x) changes from negative to positive as x passes …