Implicit Differentiation Calculator

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Implicit differentiation calculator with steps

Implicit differentiation calculator is an online tool used to differentiate the implicit function with respect to its independent variable. This implicit differentiation solver calculates the given implicit function with steps.

How does implicit derivative calculator work?

Follow the below steps to use our implicit differentiation calculator.

  • Input the f(x, y) or write the L.H.S of the implicit equation.
  • Then input the g(x, y) or write the R.H.S of the implicit equation.
  • Hit the load examples button to use the sample examples.
  • Choose the independent variable of the function i.e., x, y, or z.
  • Press the calculate key to differentiate the given implicit function.
  • Hit the possible intermediate steps to view the solution with steps.
  • Press the clear button to recalculate.

What is implicit differentiation?

Implicit differentiation helps us to differentiate the implicit equations that have both x & y variables. This technique allows us to find the slopes of tangent lines passing through curves that are not considered functions.

In explicit differentiation, equations of multivariable do not define y as a function of x so implicit differentiation is used. Explicit functions are those functions that are written in terms of independent variables only.

While the implicit functions are written in terms of both dependent and independent variables. In this type of derivative, y as a function of x doesn’t consider a constant e.g., the differential of y with respect to x will be dy/dx.

The implicit function can be written as:

\(f\left(x,y\right)=g\left(x,y\right)\)

How to differentiate the implicit function?

Follow the below example to learn how to differentiate the implicit function.

Example

Differentiate the implicit function \(4x^3+5y^2=2x^2\) with respect to “x” by using the chain rule.

Solution

Step 1: Apply the derivative notation on the given implicit equation.

\(\frac{d}{dx}\left(4x^3+5y^2\right)=\frac{d}{dx}\left(2x^2\right)\)

Step 2: Use the sum rule of differentiation and apply the derivative notation separately.

\(\frac{d}{dx}\left(4x^3\right)+\frac{d}{dx}\left(5y^2\right)=\frac{d}{dx}\left(2x^2\right)\)

Step 3: Now use the constant function and power rules to differentiate.

\(4\frac{d}{dx}\left(x^3\right)+5\frac{d}{dx}\left(y^2\right)=2\frac{d}{dx}\left(x^2\right)\)

\(4\left(3x^2\right)+5\frac{d}{dx}\left(y^2\right)=2\frac{d}{dx}\left(x^2\right)\)

\(12x^2+5\frac{d}{dx}\left(y^2\right)=2\frac{d}{dx}\left(x^2\right)\)

Step 3: Use the chain rule.

\(\frac{d}{dx}\left(y^2\right)=\frac{du^2}{du}\frac{du}{dx}where\:u=y\:\)

\(12x^2+5\left(2y\frac{d}{dx}\left(y\right)\right)=\frac{d}{dx}\left(2x^2\right)\)

\(12x^2+10y\left(\frac{d}{dx}\left(y\right)\right)=\frac{d}{dx}\left(2x^2\right)\)

Step 5: Use the chain rule again.

\(12x^2+\left(\frac{d}{dx}\left(x\right)\right)y'\left(x\right)10y=\frac{d}{dx}\left(2x^2\right)\)

\(12x^2+\left(1\right)y'\left(x\right)10y=\frac{d}{dx}\left(2x^2\right)\)

\(12x^2+y'\left(x\right)10y=\frac{d}{dx}\left(2x^2\right)\)

Step 6: The derivative of \(2x^2\) is 4x.

\(12x^2+y'\left(x\right)10y=4x\)

\(y'\left(x\right)10y=4x-12x^2\)

\( y'\left(x\right)=\frac{4x-12x^2}{10y}\)

References

  1. What is implicit differentiation | The Story of Mathematics
  2. Example of implicit differentiation | Calculus I - implicit differentiation. (n.d.)
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