Differentiation and its Applications

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An implicit function is one where the dependent variable cannot be explicitly expressed in terms of the independent variable. Example: x² + y² = 25. Here, y cannot be easily written as y = f(x) in a s…

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Differentiating both sides with respect to x: 3x² + 3y²(dy/dx) = 3a[y + x(dy/dx)] 3x² + 3y²(dy/dx) = 3ay + 3ax(dy/dx) Rearranging: 3y²(dy/dx) - 3ax(dy/dx) = 3ay - 3x² dy/dx(3y² - 3ax) = 3ay - 3x² Ther…

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Parametric functions represent x and y in terms of a parameter t: x = g(t), y = f(t). To find dy/dx, we use the chain rule: dy/dx = (dy/dt)/(dx/dt) This formula applies when dx/dt ≠ 0. Example: If x =…

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Logarithmic differentiation is used for functions of the type [f(x)]^g(x), like x^x or ((1+x)/(1-x))^(x²). Method: 1. Take log of both sides: log y = g(x) log f(x) 2. Differentiate both sides with res…

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The second order derivative is the derivative of the first derivative. If y = f(x), then: Second derivative = d/dx(dy/dx) = d²y/dx² = y'' = f''(x) Similarly, third derivative = d³y/dx³ = y''' = f'''(x…

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Cost Function: C(x) = V(x) + k, where V(x) is variable cost for producing x units and k is fixed cost. Revenue Function: R(x) = p × x, where p is price per unit and x is number of units sold. Note: Ge…

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Marginal Cost (MC) = dC/dx = C'(x) It represents the instantaneous rate of change of cost with respect to output. Marginal Revenue (MR) = dR/dx = R'(x) It represents the instantaneous rate of change o…

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Slope of tangent at point (x₀, y₀) = [dy/dx]_(x₀,y₀) Slope of normal at point (x₀, y₀) = -1/[dy/dx]_(x₀,y₀) Note: Normal is perpendicular to tangent, so their slopes are negative reciprocals of each o…