Find Implicit Differentiation Calculator

Find dy/dx Implicitly

For an equation of the form: Ax^a + By^b + Cxy + Dx + Ey + F = 0

Results

dy/dx = Not Calculated Yet

∂f/∂x = Not Calculated Yet

∂f/∂y = Not Calculated Yet

f(x,y) = Not Calculated Yet (Should be close to 0)

For f(x,y) = Ax^a + By^b + Cxy + Dx + Ey + F = 0, the implicit derivative dy/dx is calculated as – (∂f/∂x) / (∂f/∂y), where ∂f/∂x = Aax^a-1 + Cy + D and ∂f/∂y = Bby^b-1 + Cx + E.

Bar chart of partial derivatives ∂f/∂x and ∂f/∂y

What is an Implicit Differentiation Calculator?

An implicit differentiation calculator is a tool used to find the derivative of a function defined implicitly, meaning the relationship between x and y is given by an equation like F(x, y) = 0, rather than y being explicitly written as a function of x (y = f(x)). Our implicit differentiation calculator helps you find dy/dx without needing to solve for y first.

This calculator is useful for students learning calculus, engineers, physicists, and anyone dealing with equations where variables are interlinked in a complex way. For instance, the equation of a circle x² + y² = r² defines y implicitly as a function of x. Using an implicit differentiation calculator makes finding the slope of the tangent at any point on the circle straightforward.

Common misconceptions include thinking implicit differentiation is only for circles or ellipses; it applies to any equation F(x, y) = C where y cannot be easily isolated. Our implicit differentiation calculator handles a general polynomial form.

Implicit Differentiation Formula and Mathematical Explanation

When we have an equation of the form F(x, y) = C (where C is a constant, often 0), we differentiate both sides of the equation with respect to x, remembering to treat y as a function of x (y(x)). This means whenever we differentiate a term containing y, we must apply the chain rule and multiply by dy/dx.

For our calculator's equation: Ax^a + By^b + Cxy + Dx + Ey + F = 0

Differentiating with respect to x:

d/dx (Ax^a) + d/dx (By^b) + d/dx (Cxy) + d/dx (Dx) + d/dx (Ey) + d/dx (F) = d/dx (0)

Aax^a-1 + Bby^b-1(dy/dx) + C(y + x(dy/dx)) + D + E(dy/dx) + 0 = 0

Now, we group terms with dy/dx:

(dy/dx) (Bby^b-1 + Cx + E) = – (Aax^a-1 + Cy + D)

So, dy/dx = – (Aax^a-1 + Cy + D) / (Bby^b-1 + Cx + E)

This is equivalent to dy/dx = – (∂F/∂x) / (∂F/∂y), where F(x,y) = Ax^a + By^b + Cxy + Dx + Ey + F.

The implicit differentiation calculator uses this formula.

Variables Table

Variable	Meaning	Unit	Typical Range
A, B, C, D, E, F	Coefficients and constant in the equation	Dimensionless	Real numbers
a, b	Exponents in the equation	Dimensionless	Real numbers (often integers)
x, y	Coordinates of the point	Depends on context	Real numbers
∂F/∂x	Partial derivative of F with respect to x	–	Real numbers
∂F/∂y	Partial derivative of F with respect to y	–	Real numbers
dy/dx	Derivative of y with respect to x (slope)	Depends on context	Real numbers or undefined

Table explaining variables used in the implicit differentiation formula.

Practical Examples (Real-World Use Cases)

Let's see how our implicit differentiation calculator can be used.

Example 1: The Circle

Consider the equation of a circle: x² + y² = 25. This fits our form with A=1, a=2, B=1, b=2, C=0, D=0, E=0, F=-25. We want to find the slope at the point (3, 4).

• A=1, a=2, B=1, b=2, C=0, D=0, E=0, F=-25
• x=3, y=4

Using the calculator or formula: ∂F/∂x = 2x = 2(3) = 6 ∂F/∂y = 2y = 2(4) = 8 dy/dx = – (6) / (8) = -3/4.

The slope of the tangent to the circle x² + y² = 25 at (3, 4) is -3/4. The implicit differentiation calculator would confirm this.

Example 2: A More Complex Curve

Consider the curve x³ + y³ – 6xy = 0 (Folium of Descartes). We want the slope at (3, 3). Here, A=1, a=3, B=1, b=3, C=-6, D=0, E=0, F=0.

• A=1, a=3, B=1, b=3, C=-6, D=0, E=0, F=0
• x=3, y=3

∂F/∂x = 3x² – 6y = 3(3)² – 6(3) = 27 – 18 = 9 ∂F/∂y = 3y² – 6x = 3(3)² – 6(3) = 27 – 18 = 9 dy/dx = – (9) / (9) = -1.

The slope at (3, 3) is -1. Our implicit differentiation calculator is ideal for such problems.

How to Use This Implicit Differentiation Calculator

1. Enter Coefficients and Exponents: Input the values for A, a, B, b, C, D, E, and F corresponding to your implicit equation Ax^a + By^b + Cxy + Dx + Ey + F = 0.
2. Enter the Point: Input the x and y coordinates of the point at which you want to find the derivative dy/dx.
3. Calculate: Click the "Calculate" button or just change any input field. The implicit differentiation calculator will update the results in real time.
4. Read Results: The primary result is dy/dx. Intermediate results show ∂f/∂x, ∂f/∂y, and the value of f(x,y) at the point (which should be close to zero if the point is on the curve).
5. Interpret Chart: The bar chart visually represents the magnitudes of the partial derivatives.
6. Reset: Use the "Reset" button to return to default values (a circle equation).

The value of f(x,y) is important. If it's far from zero, the point (x,y) is not on the curve defined by the coefficients, and dy/dx at that point might be less meaningful for that specific curve, although it's still the value of – (∂f/∂x) / (∂f/∂y) at that point.

Key Factors That Affect Implicit Differentiation Results

1. The form of the equation: The coefficients (A, B, C, D, E, F) and exponents (a, b) define the curve and thus its slope everywhere.
2. The point (x, y): The slope dy/dx generally changes depending on the specific point on the curve.
3. Value of ∂F/∂y: If ∂F/∂y is zero at the point, dy/dx will be undefined (vertical tangent), unless ∂F/∂x is also zero (singular point). Our implicit differentiation calculator will show "Infinity or Undefined".
4. Value of ∂F/∂x: If ∂F/∂x is zero and ∂F/∂y is not, dy/dx is zero (horizontal tangent).
5. Complexity of Terms: The presence of terms like 'xy' (C ≠ 0) links the partial derivatives more intricately.
6. Exponents a and b: These determine the power to which x and y are raised, significantly influencing the derivatives.

Understanding these factors helps in interpreting the results from the implicit differentiation calculator. Looking for a {related_keywords}[0] can provide more context.

Frequently Asked Questions (FAQ)

What is implicit differentiation?

It's a technique to find the derivative of a function defined implicitly, where y is not directly given as f(x). We differentiate the entire equation with respect to x, treating y as a function of x.

Why use an implicit differentiation calculator?

It saves time and reduces errors for complex equations. Our implicit differentiation calculator quickly finds dy/dx for a general form.

When is dy/dx undefined?

When the denominator ∂F/∂y = 0 and the numerator ∂F/∂x ≠ 0 at the given point, indicating a vertical tangent.

What if both ∂F/∂x and ∂F/∂y are zero?

This indicates a singular point on the curve, which could be a cusp, node, or isolated point. The slope might be indeterminate or have multiple values depending on the path of approach.

Can this calculator handle any implicit equation?

It handles equations of the form Ax^a + By^b + Cxy + Dx + Ey + F = 0. More complex functions (like sin(y) or e^x) would require a more advanced symbolic differentiator.

What does f(x,y) near zero mean?

It means the point (x,y) is close to or on the curve defined by f(x,y)=0. If f(x,y) is far from zero, the point is not on the curve.

How do I find the equation of the tangent line?

Once you have dy/dx (the slope, m) at (x0, y0), the tangent line equation is y – y0 = m(x – x0). You might also be interested in the {related_keywords}[1].

Is implicit differentiation related to the chain rule?

Yes, very much so. When we differentiate terms involving y with respect to x, we use the chain rule because y is treated as a function of x. Learning about the {related_keywords}[2] is helpful.