In the realm of Mathematics, there are instances when the relationship between two variables is better expressed through a third variable. This third variable is referred to as a parameter . A common example of this is when curves are represented by parameters, such as x = a and y = 2at. For these types of curves, the derivative can be determined as follows:
In the following sections, we will delve into how to solve problems related to curves defined using parameters and how their derivatives can be computed through parametric differentiation.
Diving into Higher-order Derivative in Parametric Form
It is well established that the first-order derivative in parametric equations is expressed as:
Here, t represents the parameter.
Moving forward, if we differentiate the given first derivative with respect to x, we will obtain the second order derivative as shown below:
Exploring Examples
Let's delve into a few examples to better understand first-order derivatives and higher-order derivatives.
Exploring First Order Derivative Examples
Example 1: Determine dy/dx if y = cos t and x = sin t.
Solution: Given that y = cos t and x = sin t, we have
y’(t) = -sin t
x’(t) = cos t
or
Example 2: Determine dy/dx if y = tan 3 A and x = a + sec 3 A.
Solution: Given that y = tan 3 A and x = a + sec 3 A, we have
⇒ y’(A) = 3 sec 2 A.tan A.sec A
And x’(A) = 3 sec 3 A . tan 2 A
= sec A
Diving into Higher Order Derivative Examples
Example 3: Find the 2nd order derivative if x = p(A + sin A) and y = p(1 + cos A).
Solution: Given that x = p(A + sin A) and y = p(1 + cos A), we have
⇒ y’(A) = dy/dA = p cos A
⇒ x’(A) = dx/dA = p(1+sin A)
Now,
and, dx/dA= p(1+sin A)
Example 4: Find the second order derivative of the Y with respect to X, Y = (1 + t 2 )/t and x = 2t + t 2 .
Solution: Given that Y = (1 + t 2 )/t and x = 2t + t 2 , we have
The derivative of a function is the rate of change of a function with respect to a point lying in its domain.
How do you denote the first-order and second-order derivative of a function f(x)?
The first-order derivative of f(x) is denoted by f’(x), and the second-order derivative of f(x) is denoted by f’’(x).
Give the equation for the derivative of the first order in parametric form.
The derivative of the first order in parametric form is given by dy/dx = (dy/dt) × (dt/dx), where t is the parameter.
Give the equation for the derivative of the second order in parametric form.
The derivative of the second order in parametric form is given by d^2y/dx^2 = (d/dx) (dy/dx) = (d/dt)((dy/dt) × (dt/dx))× (dt/dx), where t is the parameter.