Implicit Differentiation Calculator

Find dy/dx for implicit equations involving x and y.

Supports products, quotients, powers, parentheses, trigonometric functions, logarithms, exponentials, chain rule, product rule, tangent lines and second derivatives.

Calculate an Implicit Derivative

Enter an equation such as x^2 + y^2 = 25, xy + y^2 = 10, or sin(x) + cos(y) = 1.

You can write 2x, xy, 3(x+y), or use * explicitly.

Supported Functions and Operations

Arithmetic

Addition, subtraction, multiplication, division and powers.

+ - * / ^

Implicit Multiplication

Multiplication symbols are optional in common expressions.

2x xy 3(x+y)

Trigonometry

Standard and reciprocal trigonometric functions.

sin cos tan cot sec csc

Inverse Trigonometry

Inverse trigonometric functions are supported.

asin acos atan

Exponential Functions

Exponential and logarithmic expressions can be differentiated using the chain rule.

exp(x) ln(x) log(x)

Special Functions

Additional functions for common calculus expressions.

sqrt(x) abs(x) sinh(x) cosh(x) tanh(x)

Examples of Implicit Differentiation

Example 1: Circle

Consider:

$$ x^2+y^2=25 $$

Writing the equation as \(F(x,y)=x^2+y^2-25\), we obtain:

$$ F_x=2x,\qquad F_y=2y $$

Therefore:

$$ \frac{dy}{dx} = -\frac{x}{y} $$

Example 2: Product of x and y

Consider:

$$ xy+y^2=10 $$

The product \(xy\) requires the product rule. The resulting derivative is:

$$ \frac{dy}{dx} = -\frac{y}{x+2y} $$

Example 3: Trigonometric equation

Consider:

$$ \sin(x)+\cos(y)=1 $$

Differentiating with respect to x requires the chain rule for the term involving y:

$$ \cos(x)-\sin(y)\frac{dy}{dx}=0 $$

Hence:

$$ \frac{dy}{dx} = \frac{\cos(x)}{\sin(y)} $$

How Implicit Differentiation Works

In ordinary differentiation, a function is often written explicitly as \(y=f(x)\). However, many mathematical relationships are given as equations containing both \(x\) and \(y\).

For example:

$$ x^2+y^2=25 $$

In this situation, y does not have to be isolated before differentiation. Instead, the equation can be treated as an implicit relationship between x and y.

The F(x,y) Method

Rewrite the equation as:

$$ F(x,y)=0 $$

Differentiating along the curve gives:

$$ F_x+F_y\frac{dy}{dx}=0 $$

Solving for the derivative gives the important formula:

$$ \boxed{ \frac{dy}{dx} = -\frac{F_x}{F_y} } $$

This formula is used by this calculator after constructing the symbolic expression \(F(x,y)\).

Differentiation Rules Used

Product Rule

$$ (uv)'=u'v+uv' $$

Quotient Rule

$$ \left(\frac{u}{v}\right)' = \frac{u'v-uv'}{v^2} $$

Chain Rule

$$ \frac{d}{dx}f(g(x)) = f'(g(x))g'(x) $$

Power Rule

$$ \frac{d}{dx}x^n = nx^{n-1} $$

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Frequently Asked Questions

What is implicit differentiation?

Implicit differentiation is a method for finding \(dy/dx\) when x and y are related by an equation instead of y being explicitly isolated as a function of x.

What is the formula for implicit differentiation?

If an equation can be written as \(F(x,y)=0\), then, where \(F_y\neq0\):

$$ \frac{dy}{dx} = -\frac{F_x}{F_y} $$

Why does y produce a dy/dx term?

When differentiating with respect to x, y is generally considered to depend on x. Therefore:

$$ \frac{d}{dx}y = \frac{dy}{dx} $$

Can this calculator differentiate xy?

Yes. Expressions such as xy, 2xy, and x(y+1) are interpreted using multiplication and differentiated using the product rule.

Can I calculate the slope at a particular point?

Yes. Enter both an x-coordinate and a y-coordinate. The calculator checks whether the point lies on the implicit curve and then evaluates \(dy/dx\) there.

Can this calculator find a tangent line?

Yes. When a point is supplied, the calculator evaluates the derivative at that point and uses \(y-y_1=m(x-x_1)\) to construct the tangent line.

What happens if Fy is zero?

The formula \(dy/dx=-F_x/F_y\) requires \(F_y\neq0\). If \(F_y=0\), the usual local representation of the curve as y as a function of x may fail at that point. The curve can instead have a vertical tangent or require a different local analysis.