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Parametric Derivative Calculator + Online Solver With Free Steps
The Parametric Derivative Calculator is used to compute the derivative of the parametric functions. The calculator takes the parametric functions x and y as input. They can be a function of t or any other variable.
Usually, t is taken as the parameter in the parametric equations. The parametric equations have the dependent variables x and y and the independent variable t. The dependent variables are continuous functions of the independent variable.
The parametric equation helps in removing the dependency of variable y on the variable x. It introduces another variable, such as t, on which both the variables are now dependent.
The calculator computes the first derivative by differentiating the parametric functions x and y to t separately. The first derivatives initially calculated by the calculator are dy/dt and dx/dt.
The first derivative required in the result is dy/dx. Dividing the numerator and denominator in dy/dx by dt gives the numerator as dy/dt and the denominator as dx/dt.
The calculator places the values in the numerator and denominator and simplifies to get the required first derivative of the parametric functions.
What Is a Parametric Derivative Calculator?
The Parametric Derivative Calculator is an online tool used to calculate the first derivative of the parametric equations x and y, which contains a parameter t or any other variable entered by the user.
The calculator requires both the functions x(t) and y(t) to compute the parametric first derivative function.
The parametric derivative is the derivative between two dependent variables, x and y. It takes their derivatives separately with respect to the independent variable, such as time t or any other variable.
The parametric first derivative helps find the tangent line equation on the parametric curve formed by the two parametric equations.
How To Use the Parametric Derivative Calculator
The user can use the Parametric Derivative Calculator by following the steps given below.
Step 1
The user must first enter the first parametric equation in the calculator’s input tab. It should be entered in the block labeled, “Given x =.”
The parametric equation should have an independent variable which can be time t or any other parameter but not x or y, as they are the dependent variables.
For the default example, the first parametric function x(t) is taken as 2t-3.
Step 2
The user must now enter the second parametric function in the input tab of the calculator. It should be entered in the block titled “and y=.”
The parametric function can be a function of t or any other variable according to the user’s requirement. But it should be the same as used in the first parametric function of x.
The parametric function for y is taken as $3t^2 \ – \ 2t$ in the default example.
Step 3
The user must now enter the parameter or the independent variable used in both parametric equations. Usually, time t is taken as the parameter in the parametric functions.
But the calculator provides the option to use a different parameter if the user wants. The parameter should be entered in the block labeled against “The first derivative with respect to.”
In the default example, as both x and y functions are in the variable t, t is entered in this input block.
Step 4
As both the functions x(t) and y(t) and the variable t are entered in the calculator’s input window, the user should now press “Calculate” for the calculator to process the input data.
Output
The Parametric Derivative Calculator computes the output and displays it in the two windows below.
Input
The calculator interprets the input and displays the parametric functions x and y in this window. It displays the functions by placing them in the mathematical formula for the parametric first derivative. The formula for the parametric derivative is given as:
\[ \frac{dy}{dx} = \frac{ \frac{dy}{dt} }{ \frac{dx}{dt} } \]
For the default example, the calculator shows the Input functions as follows:
\[ \frac{dy}{dx} = \frac{ \frac{ d(3t^2 \ – \ 2t) }{dt} }{ \frac{ d(2t \ – \ 3) }{dt} } \]
Result
The calculator calculates the first derivative of the parametric equations and shows the final result in this window. The mathematical steps for the default example are as follows:
Calculating dy/dt gives:
\[ \frac{dy}{dt} = \frac{ d(3t^2 \ – \ 2t) }{dt} = 3(2t) \ – \ 2 = 6t \ – \ 2 \]
Computing dx/dt gives:
\[ \frac{dx}{dt} = \frac{ d(2t \ – \ 3) }{dt} = 2 \]
Putting the values for dy/dt and dx/dt, the calculator gives the final result as follows:
\[ \frac{dy}{dx} = \frac{ \frac{dy}{dt} }{ \frac{dx}{dt} } = \frac{1}{2} (6t \ – \ 2) \]
Solved Examples
The following examples are solved through the Parametric Derivative Calculator.
Example 1
For the parametric functions:
\[ x(t) = t^3 \]
\[ y(t) = 6t^2 \ – \ 15t^3 \]
Calculate the first derivative of the parametric equations.
Solution
The user must first enter the parametric equations x(t) and y(t) in the calculator’s input window, as given in the example.
The parameter t is used, so t should be entered in the “The first derivative with respect to” block.
After pressing “Calculate,” the calculator first displays the parametric input functions by putting them in the formula for the derivative of parametric equations.
The Input window shows the following equation:
\[ \frac{dy}{dx} = \frac{ \frac{ d( 6t^2 \ – \ 15t^3 ) }{dt} }{ \frac{ d( t^3 ) }{dt} } \]
The calculator computes the first derivative and shows the simplified result as follows:
\[ \frac{dy}{dx} = \frac{ 12t \ – \ 45t^2 }{ 3t^2 } \]
Example 2
Calculate the first derivative of the parametric equations x(t)=t and y(t)=7$t^2$.
Solution
The parametric functions x and y and the parameter t should be entered in the calculator’s input window. The calculator shows the Input parametric functions in the output window as follows:
\[ \frac{dy}{dx} = \frac{ \frac{ d( 7t^2 ) }{dt} }{ \frac{ d( t ) }{dt} } \]
The calculator shows the final result of the first derivative as follows:
\[ \frac{dy}{dx} = 14t \]