Differentiation of Parametric Functions¶
Differentiation of parametric functions involves finding the derivative of a function that is defined in terms of a parameter, rather than as an explicit function of \(x\). In parametric differentiation, the coordinates \(x\) and \(y\) are given as functions of a third variable, typically \(t\), called the parameter.
If \(x\) and \(y\) are given as: [ x = f(t) \quad \text{and} \quad y = g(t), ] where both \(x\) and \(y\) are expressed in terms of the parameter \(t\), then the derivative of \(y\) with respect to \(x\), denoted as \(\frac{dy}{dx}\), can be found using the following steps:
Formula for Parametric Differentiation¶
To differentiate parametric functions, use the formula: [ \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}, ] where: - \(\frac{dy}{dt}\) is the derivative of \(y\) with respect to \(t\). - \(\frac{dx}{dt}\) is the derivative of \(x\) with respect to \(t\).
Steps for Differentiating Parametric Functions¶
- Differentiate \(x\) with respect to \(t\): Find \(\frac{dx}{dt}\).
- Differentiate \(y\) with respect to \(t\): Find \(\frac{dy}{dt}\).
- Divide \(\frac{dy}{dt}\) by \(\frac{dx}{dt}\): This gives \(\frac{dy}{dx}\), the derivative of \(y\) with respect to \(x\).
Example 1: Parametric Differentiation¶
Given the parametric equations: [ x = t^2 + 2t, \quad y = 3t - 4, ] find \(\frac{dy}{dx}\).
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Differentiate \(x\) with respect to \(t\): [ \frac{dx}{dt} = 2t + 2. ]
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Differentiate \(y\) with respect to \(t\): [ \frac{dy}{dt} = 3. ]
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Divide \(\frac{dy}{dt}\) by \(\frac{dx}{dt}\): [ \frac{dy}{dx} = \frac{3}{2t + 2}. ]
Example 2: Finding the Second Derivative¶
To find the second derivative \(\frac{d^2y}{dx^2}\), first find the derivative of \(\frac{dy}{dx}\) with respect to \(t\) and then divide by \(\frac{dx}{dt}\): [ \frac{d^2y}{dx^2} = \frac{\frac{d}{dt}\left(\frac{dy}{dx}\right)}{\frac{dx}{dt}}. ]
Using the previous example: 1. Differentiate \(\frac{dy}{dx} = \frac{3}{2t + 2}\) with respect to \(t\): [ \frac{d}{dt}\left(\frac{3}{2t + 2}\right) = \frac{-6}{(2t + 2)^2}. ]
- Divide by \(\frac{dx}{dt}\): [ \frac{d^2y}{dx^2} = \frac{\frac{-6}{(2t + 2)^2}}{2t + 2} = \frac{-6}{(2t + 2)^3}. ]
Example 3: Parametric Equations Involving Trigonometric Functions¶
Given: [ x = \cos(t), \quad y = \sin(t), ] find \(\frac{dy}{dx}\).
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Differentiate \(x\) with respect to \(t\): [ \frac{dx}{dt} = -\sin(t). ]
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Differentiate \(y\) with respect to \(t\): [ \frac{dy}{dt} = \cos(t). ]
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Divide \(\frac{dy}{dt}\) by \(\frac{dx}{dt}\): [ \frac{dy}{dx} = \frac{\cos(t)}{-\sin(t)} = -\cot(t). ]
Parametric differentiation is especially useful in physics and engineering, where variables are often described in terms of a parameter, such as time. This method allows for analyzing the rate of change and curvature of parametric curves.
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