Introduction to the Parametric Differentiation
In mathematics, the differentiation of parametric equations is a process of calculating the rate of change of parametric equations. This type of differentiation is used when an equation is defined as a relation with another parameter. Let’s understand what is the derivative of parametric equations and how to calculate it by using derivatives rules and formulas.
Understanding of the Parametric Differentiation
In calculus, some functions of one or two variables x and y, are defined so complicated that we need to define another variable as a function of both variables. This another variable that we use to write the function in simple form is known as a parameter and such an equation is called parametric equation. Moreover, finding the rate of change of parametric equations is called parametric differentiation.
Derivative of a parametric equation can be calculated by using differentiation rules which allows us to find the rate of change in the function. An example of a parametric equation is, x = cos t and y = sin t.
Formula of Parametric Differentiation
To find derivatives of a parametric equation, we use the chain rule of the derivative. Sometimes, we need to use the following rules of derivatives of the chain rule.
- Chain Rule:
If two functions are in a combination form with each other, the derivative of these functions can be written as;
\[ \frac{d}{dx}\left[ f(g(x)) \right] = \frac{dy}{du} \cdot \frac{du}{dx} \]
Where
\( u = g(x) \) and \( y = f(u) \)
- Product Rule:
If two functions are multiplied with each other, the derivative of these functions can be written as;
\[ \frac{d}{dx}\left[ f(x)g(x) \right] = f'(x)g(x) + g'(x)f(x) \]
Where two functions are,
\( f(x) \) and \( g(x) \)
- Power Rule:
The power rule is a rule to calculate derivatives of an algebraic expression with some power. The formula for power rule for xn is;
\[ f'(x) = \frac{d}{dx}\left[ x^n \right] = n \cdot x^{n-1} \cdot \frac{d}{dx}(x) \]
Where, n is a real number.
- Quotient Rule
If two functions, f(x) and g(x), are in fractional form, the derivative of quotient of functions can be written as;
\[ \frac{d}{dx}\left[ \frac{f(x)}{g(x)} \right] = \frac{f(x)g'(x) - f'(x)g(x)}{\left[ g(x) \right]^2} \]
- Constant Multiple Rule
If a constant a is multiplied with a function f(x), then the constant multiple rule can be written as;
\[ \frac{d}{dx}\left[ a \cdot f(x) \right] = a \cdot f'(x) \]
How do you calculate derivatives of parametric equations?
The calculations of the parametric equation differentiation is quite different from finding derivatives of an algebraic or trigonometric function. Here, we provide you a step-by-step procedure to differentiate parametric equations. Follow these steps;
- Identify the function f(x, y). If both variables x and y depend on the third variable i.e. t, the function will be a parametric function.
- We need to calculate the derivative as dy/dx. For this, first calculate the derivative of x=f(t) and y=g(t) with respect to t.
- Now use the above derivatives in the chain rule formula to get dy/dx.
- Simplify if needed.
Let’s understand the calculations of parametric differentiation in the following example.
Derivative of parametric equation example
Suppose that we want to calculate dy/dx we,
$x= \cos t$
$y=\sin t$
Let’s calculate the derivative of x and y with respect to t, that is;
$\frac{dx}{dt}=-\sin t$
And
$\frac{dy}{dt}=\cos t$
Now by using the chain rule calculator,
$\frac{dy}{dx}= \frac{dy}{dt} \times \frac{dt}{dx}$
Substituting the values of dy/dt and dt/dx,
$\frac{dy}{dx}=-\sin t \times \frac{1}{\cos t}$
Since,
\[ \frac{\sin t}{\cos t} = \tan t \]
Then,
$\frac{dy}{dx}=-\tan t$
Hence the derivative of the given function is -tan t.
Conclusion
In calculus, parametric differentiation is a process of finding derivatives of a parametric equation i.e, the equations whose variables depend on a third parameter. It has many applications in calculus and trigonometry and is used to calculate the rate of change of different curves in two or three-dimensions. This derivative can be calculated by using differentiation chain rule.