This calculator efficiently determines the slope of the tangent line ($dy/dx$) at a specific point on an implicitly defined curve, such as $x^3 + y^3 = 6xy$. Implicit differentiation is crucial when $y$ cannot be easily isolated as a function of $x$.
Implicit Derivative Calculator
Result ($dy/dx$):
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Implicit Derivative Formula
The general procedure for implicit differentiation:
- Differentiate both sides of the equation with respect to $x$. Treat $y$ as a function of $x$.
- Apply the chain rule to any term involving $y$, multiplying by $\frac{dy}{dx}$.
- Isolate the terms containing $\frac{dy}{dx}$ on one side.
- Factor out $\frac{dy}{dx}$ and solve for the derivative.
$$ \frac{dy}{dx} = \frac{2y – x^2}{y^2 – 2x} $$
Formula Source (Implicit Differentiation): LibreTexts Math – Implicit Differentiation, Paul’s Online Math Notes
Variables
- Equation: The algebraic relationship between $x$ and $y$. This calculator uses the Folium of Descartes ($x^3 + y^3 = 6xy$) as a running example.
- X-Value: The $x$-coordinate of the point on the curve where the slope is being evaluated.
- Y-Value: The $y$-coordinate of the point on the curve where the slope is being evaluated.
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What is Implicit Differentiation?
Implicit differentiation is a technique used in calculus to find the derivative of implicitly defined functions. A function is defined implicitly when the dependent variable ($y$) is not isolated on one side of the equation. Standard differentiation rules (like the power rule and product rule) are usually applied to explicit functions, but for complex, interwoven equations like $x^2y + y^3 = 10$, a different approach is needed.
The fundamental idea is to differentiate both sides of the equation with respect to $x$, treating $y$ as an unknown, differentiable function of $x$. Every time a term involving $y$ is differentiated, the Chain Rule dictates that the result must be multiplied by $dy/dx$. The result, $dy/dx$, gives the slope of the tangent line to the curve at any point $(x, y)$.
How to Calculate Implicit Derivative (Example)
Using the example equation $x^3 + y^3 = 6xy$, we will find $\frac{dy}{dx}$ at the point $(3, 3)$.
- Differentiate the Equation: Differentiate both sides with respect to $x$: $$ \frac{d}{dx} (x^3 + y^3) = \frac{d}{dx} (6xy) $$ $$ 3x^2 + 3y^2 \frac{dy}{dx} = 6y \cdot \frac{d}{dx}(x) + 6x \cdot \frac{d}{dx}(y) $$ $$ 3x^2 + 3y^2 \frac{dy}{dx} = 6y(1) + 6x \frac{dy}{dx} $$
- Isolate $\frac{dy}{dx}$ Terms: Gather all $\frac{dy}{dx}$ terms on the left side: $$ 3y^2 \frac{dy}{dx} – 6x \frac{dy}{dx} = 6y – 3x^2 $$
- Solve for $\frac{dy}{dx}$: Factor out the derivative and divide: $$ \frac{dy}{dx} (3y^2 – 6x) = 6y – 3x^2 $$ $$ \frac{dy}{dx} = \frac{6y – 3x^2}{3y^2 – 6x} = \frac{2y – x^2}{y^2 – 2x} $$
- Evaluate at $(3, 3)$: Substitute $x=3$ and $y=3$ into the final expression: $$ \frac{dy}{dx} \Big|_{(3, 3)} = \frac{2(3) – (3)^2}{(3)^2 – 2(3)} = \frac{6 – 9}{9 – 6} = \frac{-3}{3} = -1 $$
Frequently Asked Questions (FAQ)
What is the difference between explicit and implicit differentiation?
Explicit differentiation is used when the function is written as $y = f(x)$, meaning $y$ is explicitly defined in terms of $x$. Implicit differentiation is used when the variables are mixed, such as $x^2 + y^2 = r^2$, and $y$ cannot be easily or cleanly solved in terms of $x$.
When must I use the Chain Rule in implicit differentiation?
The Chain Rule must be used every time you differentiate a term containing the variable $y$. Since $y$ is treated as a function of $x$ (i.e., $y(x)$), differentiating $\frac{d}{dx}[y^n]$ becomes $n y^{n-1} \cdot \frac{dy}{dx}$.
Can I use implicit differentiation on functions that are also explicit?
Yes. If $y = x^2$, explicit differentiation gives $dy/dx = 2x$. Implicit differentiation gives: $\frac{d}{dx}(y) = \frac{d}{dx}(x^2) \Rightarrow \frac{dy}{dx} = 2x$. The results are identical, but the implicit method is often overkill for explicit functions.
What does the result of the implicit derivative represent?
The result, $dy/dx$, represents the instantaneous rate of change of $y$ with respect to $x$, which is geometrically the slope of the tangent line to the implicit curve at the specific point $(x, y)$.