Lmplicit Differentiation Calculator
Lmplicit Differentiation - Solve mathematical problems with step-by-step solutions.
Implicit Differentiation Calculator
Find dy/dx for implicit functions
Implicit Differentiation
Implicit differentiation is a method used to find the derivative of a dependent variable in an implicit function, by differentiating each term with respect to the independent variable. This calculator finds dy/dx using the formula: dy/dx = - (∂F/∂x) / (∂F/∂y), where the equation is in the form F(x, y) = 0.
How the Implicit Differentiation Calculator Works
Implicit differentiation allows you to find derivatives when y is not explicitly solved in terms of x. Instead of y = f(x), you have an equation like x² + y² = 25. You differentiate both sides with respect to x, treating y as a function of x.
Explicit vs Implicit
In explicit form, y is isolated (like y = 3x² + 5), and you can differentiate directly. In implicit form, y is mixed with x (like x² + y² = 25), requiring you to use the chain rule with dy/dx when differentiating y terms.
The Process
Step 1: Differentiate Both Sides
Take d/dx of the entire equation. Apply differentiation rules to every term, remembering that d/dx applies to both sides of the equals sign.
Step 2: Apply Chain Rule to y Terms
Every time you differentiate a term containing y, multiply by dy/dx using the chain rule. For example, d/dx(y²) = 2y × dy/dx. This is because y is implicitly a function of x.
Step 3: Collect dy/dx Terms
Move all terms containing dy/dx to one side of the equation and all other terms to the opposite side. This prepares you to solve for dy/dx.
Step 4: Solve for dy/dx
Factor out dy/dx and isolate it by dividing both sides. Your final answer may contain both x and y variables, which is perfectly normal for implicit differentiation.
Example: x² + y² = 25
Step 1: Differentiate Both Sides
d/dx(x² + y²) = d/dx(25)
Step 2: Apply Rules
2x + 2y(dy/dx) = 0
Note how differentiating y² gives us 2y × dy/dx (chain rule)
Step 3: Collect dy/dx Terms
2y(dy/dx) = -2x
Step 4: Solve
dy/dx = -2x / 2y = -x/y
Common Patterns
Power of y
d/dx(yn) = n × y(n-1) × (dy/dx)
Example: d/dx(y³) = 3y² × dy/dx
Product with y
d/dx(xy) = x(dy/dx) + y (product rule)
First times derivative of second + second times derivative of first
Function of y
d/dx(sin(y)) = cos(y) × (dy/dx)
Chain rule: derivative of outer × derivative of inner
Frequently Asked Questions
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