Introduction to Implicit Differentiation

An implicit function is a type of function in which both dependent and independent variables are in relation to each other. The concept of implicit differentiation is used to find the derivative of an implicit function. The implicit differentiation in calculus is a fundamental way to find the rate of change of implicit expressions. Let’s learn more about implicit differentiation and understand how to apply the implicit differentiation formula.

Understanding the Implicit Differentiation

Since the derivative is the rate of change of a function with respect to an independent variable, this rate of change is also known as the slope of the tangent line, which is calculated simply by using different derivative rules. But when a function is not expressed as an explicit function, i.e., y=f(x), we cannot use ordinary derivative rules to find the derivative. For this, implicit differentiation is used. 

The implicit derivative of a function is defined as;

“The derivative calculated by differentiating the implicit function with respect to an independent variable and treating other variables as constant is known as implicit differentiation. It is expressed by simply dy/dx.”

Implicit Differentiation Formula

Although there is no specific formula for implicit differentiation because only rearrangement is required to find the derivative, this formula is derived from calculating the derivative of an implicit function which is a function that is just an expression. In an implicit function, the dependent and independent variables are combined. 

For example, the implicit derivative of a function xy=1 is calculated as;

$\frac{d}{dx}(xy)=\frac{d}{dx}(1)$

Since the derivative of a constant number is zero. Therefore $\frac{d}{dx}(1) = 0$. Using product rule of derivative on the left side, 

$x.\frac{dy}{dx}+y(1)=0$

By rearranging above equation, 

$\frac{dy}{dx} = -\frac{y}{x}$

The implicit differentiation helps to calculate derivative of inverse trigonometric functions and makes the implicit equations easier to solve. 

How to apply an implicit differentiation formula?

The implementation of implicit differentiation is divided into a few steps. These steps assist us to calculate the derivative of an implicit function. These steps are:

1. Write the expression of the function. 
2. Identify the independent and dependent variables. 
3. Apply the derivative on both sides of the equation with respect to the independent variable. 
4. Calculate the derivative one-by-one by using derivative rules. For example if the left side of the equation contains a power function, use the power rule derivative formula.
5. In the final step, simplify the equation and rearrange it to get dy/dx.

Let’s understand the following examples by applying implicit differentiation.

Implicit Differentiation example 1

Since the implicit differentiation helps to calculate the derivative of trigonometric inverse functions. So, we can use it to calculate the cos inverse derivative. For this suppose that,

$y = \cos^{-1}x$

It can be written as;

$\cos y = x$

Now the function bechttps://calculator-derivative.com/inverse-trigonometric-differentiationomes an implicit function. So, differentiating both sides,

$\frac{d}{dx}[\cos y] = \frac{d}{dx}[x]$

Since the derivative of tan y is –sin y and the derivative of x is 1. Then, 

$-\sin y\frac{dy}{dx} = 1$

Solving the equation to find dy/dx, we get

$\frac{dy}{dx} = -\frac{1}{\sin y}$______(A)

By using trigonometric ratios, we know that

$\sin^2y + \cos^2y = 1$

Since $x = \cos y$, 

$\sin^2y + x^2 = 1$

Or, 

$\sin^2y = 1 – x^2$

Taking square roots on both sides.

$\sin y = \sqrt{1-x^2}$

Now substituting the value of sin y in the equation A.

$\sqrt{1-x^2}dy/dx = 1$

Rearranging the equation to get dy/dx.

$\frac{dy}{dx}=-\frac{1}{\sqrt{1-x^2}}$

Hence the above equation gives the derivative of cos inverse x. You can also use the implicit differentiation calculator to calculate derivative of inverse functions. 

Implicit Differentiation example 2

To calculate the derivative of sin inverse, we will use the following steps.

$y = \sin^{-1}x$

It can be written as;

$\sin y = x$

Now the function becomes an implicit function. So, differentiating both sides,

$\frac{d}{dx}[\sin y] = \frac{d}{dx}[x]$

Since the derivative of sin y is cos y and the derivative of x is 1. Then, 

$\cos y\frac{dy}{dx} = 1$

Solving the equation to find dy/dx, we get

$\frac{dy}{dx} = \frac{1}{\cos y}$ ____(A)

By using trigonometric ratios, we know that

$\sin^2y + \cos^2y = 1$

Since $x = \sin y$, 

$\cos^2y + x^2 = 1$

Or, 

$\cos^2y = 1 – x^2$

Taking square roots on both sides.

$\cos y=\sqrt{1-x^2}$

Now substituting the value of cos y in the equation A.

$\sqrt{1-x^2}\frac{dy}{dx} = 1$

Rearranging the equation to get dy/dx.

$\frac{dy}{dx} = \frac{1}{\sqrt{1-x^2}}$

Above equation is the derivative of arcsin by using implicit differentiation.

Applying Implicit Differentiation formula by using calculator

The derivative of an implicit function can be also calculated by using a differential calculator. It is an online tool that follows the derivative formula to evaluate derivative of a function. You can find it online by searching for a derivative calculator. For example, to calculate the derivative of tan inverse, the following steps are used by using this calculator.

1. Write the expression of the power function in the input box such as, tan^-1(x).
2. Choose the variable to calculate the rate of change, which will be x in this example.
3. Review the input so that there will be no syntax error in the function. 
4. Now at the last step, click on the calculate button. By using this step, the implicit function calculator will provide the derivative of arctan quickly and accurately which will be $\frac{1}{1+x^2}$.

Comparison between Implicit differentiation and partial derivative 

The comparison between the implicit differentiation and partial derivative can be easily analysed using the following difference table.

Conclusion

Implicit differentiation is a fundamental concept of derivatives that are used to differentiate an implicit function. But there is no specific formula to calculate the implicit derivative. We can calculate it by differentiating the implicit expression and then solving it to find dy/dx.