Implicit differentiation calculator is used to find the differential of implicit function with respect to its variable. This implicit derivative calculator evaluates the implicit equation step-by-step.
The implicit differentiation solver is a type of differential calculator.
Follow the steps below to solve the problems of implicit function.
Implicit differentiation is a process of finding the differential of a dependent variable in an implicit function by expressing the differential of the dependent variable as a symbol and by differentiating each term separately.
Implicit differentiation calculates the dy/dx of the given equation. This type of differentiation solves the equations without taking y as a constant w.r.t “x”. For example, the implicit differential of \(y^2\) w.r.t “x” is \(2y\frac{dy}{dx}\).
The general equation of implicit is:
\(f\left(x,y\right)=g\left(x,y\right)\)
To calculate the implicit differentiation of the equation, we have to apply the differential on both sides of the equation. dy/dx calculator provides the accurate result of the implicit function.
Following are a few examples solved by our implicit differentiation solver.
Example 1
Calculate the implicit differentiation of \(3x^2y+4y^2=23\) w.r.t “x”.
Solution
Step 1: Use the differentiation notation in the given implicit function.
\( \frac{d}{dx}\left(3x^2y+4y^2\right)=\frac{d}{dx}\left(23\right)\)
Step 2: Differentiate each term separately and apply the power, product, and constant rules.
\(\frac{d}{dx}\left(3x^2y\right)+\frac{d}{dx}\left(4y^2\right)=\frac{d}{dx}\left(23\right)\)
\(y\frac{d}{dx}\left(3x^2\right)+3x^2\frac{d}{dx}\left(y\right)+\frac{d}{dx}\left(4y^2\right)=\frac{d}{dx}\left(23\right)\)
\( y\left(3\cdot 2x^{2-1}\right)+3x^2\frac{d}{dx}\left(y\right)+\left(4\cdot 2y^{2-1}\frac{d}{dx}\left(y\right)\right)=0\)
\(6xy+3x^2\frac{dy}{dx}+8y\frac{dy}{dx}=0\)
Step 3: Now separate the dy/dx term.
\(3x^2\frac{dy}{dx}+8y\frac{dy}{dx}=-6xy\)
\(\left(3x^2+8y\right)\frac{dy}{dx}=-6xy\)
\(\frac{dy}{dx}=-\frac{6xy}{\left(3x^2+8y\right)}\)
Example 2
Calculate the implicit differentiation of \(3x^2+6y^2=3x\) w.r.t “x” by using the chain rule.
Solution
Step 1: Use the differentiation notation in the given implicit function.
\(\frac{d}{dx}\left(3x^2+6y^2\right)=\frac{d}{dx}\left(3x\right)\)
Step 2: Now differentiate the above terms separately and apply the power rule.
\(\frac{d}{dx}\left(3x^2\right)+\frac{d}{dx}\left(6y^2\right)=\frac{d}{dx}\left(3x\right)\)
\(3\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(6y^2\right)=\frac{d}{dx}\left(3x\right)\)
\(3\left(2x\right)+\frac{d}{dx}\left(6y^2\right)=\frac{d}{dx}\left(3x\right)\)
\(6x+\frac{d}{dx}\left(6y^2\right)=\frac{d}{dx}\left(3x\right)\)
Step 3: Apply the chain rule.
\(\frac{d}{dx}\left(y^2\right)=\frac{du^2}{du}\cdot \frac{du}{dx}where\:u=y\:\&\:\frac{d}{du}\left(u^2\right)=2u\)
Step 4: Simplify the expression.
\(6x+6\left(2y\frac{d}{dx}\left(y\right)\right)=\frac{d}{dx}\left(3x\right)\)
\(6x+12\left(\frac{d}{dx}\left(y\right)\right)y=\frac{d}{dx}\left(3x\right)\)
Step 5: Using chain rule
\( \frac{d}{dx}\left(y\right)=\frac{dy\left(u\right)}{dx}\cdot \frac{du}{dx},\:where\:u=x\:\&\:\frac{d}{du}\left(y\left(u\right)\right)=y'\left(u\right)\)
\(6x+\left(\frac{d}{dx}\left(x\right)\right)y'\left(x\right)12y=\frac{d}{dx}\left(3x\right)\)
Step 6: The derivative of 3x is 3.
\(6x+y'\left(x\right)12y=3\)
\(6x+12y\frac{dy}{dx}=3\)
\(12y\frac{dy}{dx}=3-6x\)
\(\frac{dy}{dx}=\frac{3-6x}{12y}\)
Here are some examples of implicit functions solved by our implicit function calculator.
| Implicit differentiation of | Result |
| xy=1 | \(\frac{-y}{x}\) |
| xy+sin(xy)=1 | \(\frac{-y}{x}\) |
| xy-x+2y=1 | \(\frac{1-y}{x+2}\) |
| sqrt(xy)=x^2y+1 | \(\frac{4xy\sqrt{xy}-y}{x\:-2x^2\sqrt{xy}}\) |
| e^xy | \(\frac{ye^{xy}}{1-xe^{xy}}\) |
