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Implicit Differentiation Calculator

Write down functions and points to calculate implicit differentiation through this implicit differentiation calculator.

Enter a function: f(x,y) = g(x,y)

f(x,y):

g(x,y):

optional W.R.T:

Evaluate the derivative at the point (x,y): (Differentiate)

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Implicit Differentiation Calculator

The Implicit Differentiation Calculator helps you find the derivative of functions expressed in implicit form. It can also evaluate the derivative at given points, allowing you to compute dy/dx efficiently without manually solving for y.

What is Implicit Differentiation?

In calculus, some functions are expressed implicitly, meaning the dependent variable y is not isolated on one side of the equation. The function involves both x and y. For example, the circle equation is implicit:

\( x^2 + y^2 = r^2 \)

Implicit differentiation finds the derivative of the dependent variable by differentiating each term of the equation, treating y as a function of x, and solving for dy/dx.

Steps to Find dy/dx by Implicit Differentiation:

Start with the given equation involving two variables.
Differentiating both sides of the equation with respect to the chosen variable.
Collect all dy/dx terms on one side and factor them out.
Solve algebraically for dy/dx.
Substitute specific values for the variables (if needed) to evaluate the derivative at a point.

You don’t need to do these steps manually—input the function into our calculator for precise results.

Example

Find the derivative of \( a^3 + b^3 = 2ab \) at the point (a, b) = (2, 3).

Solution:

Given equation:

\( a^3 + b^3 = 2ab \)

Differentiating both sides with respect to a:

\( 3a^2 + 3b^2 \frac{db}{da} = 2a \frac{db}{da} + 2b \)

Rewriting to isolate \(\frac{db}{da}\):

\( 3b^2 \frac{db}{da} - 2a \frac{db}{da} = 2b - 3a^2 \)

\( \frac{db}{da} (3b^2 - 2a) = 2b - 3a^2 \)

\( \frac{db}{da} = \frac{2b - 3a^2}{3b^2 - 2a} \)

Substitute a = 2 and b = 3:

\( \frac{db}{da} = \frac{2(3) - 3(2)^2}{3(3)^2 - 2(2)} = \frac{6 - 12}{27 - 4} = \frac{-6}{23} \)

Thus, the derivative at (2, 3) is:

\( \frac{db}{da} = -\frac{6}{23} \)

How the Implicit Differentiation Calculator Works

Input:

Enter the function \( f(x, y) = g(x, y) \).
Select the variable to differentiate with respect to.
Optionally, provide values of x and y to evaluate the derivative at a specific point.
Click Calculate to get the result.

Output:

The calculator provides the implicit derivative \( \frac{dy}{dx} \).
It can also evaluate the derivative at the specific points entered.

FAQ

Why do we use implicit differentiation?

Implicit differentiation is used when the dependent variable y is not isolated. It allows finding dy/dx without explicitly solving for y.

What is the difference between explicit and implicit functions?

An explicit function expresses y solely in terms of x. An implicit function involves both x and y, where y is not isolated.

What is the implicit differentiation of ab?

The derivative of \( ab \) is:

\( \frac{d}{dx}(ab) = a \frac{db}{dx} + b \)

Conclusion

Use this online Implicit Differentiation Calculator to quickly compute derivatives of functions where the dependent variable is not isolated. It also allows evaluation of dy/dx at specific points for practical applications.

References

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