Contents

# 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:

\[ \frac{dy}{dx} = 14t \]