Implicit differentiation lets you find slopes on curves that are not written as y = f(x). This matters because many important shapes, such as circles, ellipses, and hyperbolas, are defined by equations involving both x and y together. Instead of solving for y first, you differentiate both sides with respect to x and remember that y changes as x changes.
The result often gives dy/dx in terms of both x and y, which is perfect for finding tangent lines at specific points.
The key mechanism is the chain rule: whenever you differentiate a term involving y, multiply by dy/dx. For example, d(y^2)/dx = 2y dy/dx, not just 2y. In applications, implicit differentiation is used to find tangent slopes, normal lines, and related rates when variables are connected by an equation.
A typical workflow is to differentiate the whole equation, solve for dy/dx, substitute the point or values, then use the slope in a tangent line or rate equation.
Key Facts
- If F(x, y) = 0, then dy/dx = -F_x/F_y when F_y is not 0.
- For a circle x^2 + y^2 = r^2, 2x + 2y dy/dx = 0, so dy/dx = -x/y.
- For an ellipse x^2/a^2 + y^2/b^2 = 1, dy/dx = -b^2 x/(a^2 y).
- The tangent line at (x1, y1) with slope m is y - y1 = m(x - x1).
- The normal line slope is m_normal = -1/m_tangent when the tangent slope is nonzero.
- In related rates, differentiate with respect to time: d(x^2 + y^2)/dt = 2x dx/dt + 2y dy/dt.
Vocabulary
- Implicit equation
- An equation that defines a relationship between x and y without necessarily solving for y alone.
- Implicit differentiation
- A method for finding dy/dx by differentiating both sides of an equation while treating y as a function of x.
- Tangent line
- A line that touches a curve at a point and has the same instantaneous slope as the curve there.
- Normal line
- A line perpendicular to the tangent line at a point on a curve.
- Related rates
- Problems where changing quantities are connected by an equation and their rates of change are found by differentiating with respect to time.
Common Mistakes to Avoid
- Forgetting dy/dx after differentiating a y term is wrong because y depends on x. For example, d(y^2)/dx must be 2y dy/dx, not 2y.
- Substituting the point before differentiating can hide the variables and destroy the slope relationship. Differentiate first, solve for dy/dx, then plug in the point.
- Using the tangent slope formula without checking for vertical tangents is wrong because dy/dx may be undefined when the denominator is zero. A vertical tangent is written as x = constant, not y = mx + b.
- Treating related rates as ordinary derivatives with respect to x is wrong because the variables usually change with time. Use dx/dt, dy/dt, and dA/dt when the problem describes motion or changing quantities.
Practice Questions
- 1 Find dy/dx for the circle x^2 + y^2 = 25, then find the tangent line at the point (3, 4).
- 2 For the ellipse x^2/16 + y^2/9 = 1, find dy/dx and the slope of the tangent line at the point (2, 3sqrt(3)/2).
- 3 A point moves along x^2 + y^2 = 100. If x is increasing while y is positive, explain conceptually why y must be decreasing and how the sign of dy/dt shows this.