Radford Mathematics IB Mathematics resources • AA HL

IB Mathematics: Analysis and Approaches HL — Topic 5 Calculus

Implicit Differentiation

Differentiate relations where y is not isolated, using the chain rule on every y-term.

AA HL · AHL 5.14

Learning goal

Use implicit differentiation to find gradients, tangents, normals and stationary points.

Syllabus link

AA HL: AHL 5.14 · Topic 5 Calculus.

Big idea

When differentiating with respect to x, every function of y produces a factor dy/dx.

Key relationship

\(\frac{d}{dx}(y^n)=ny^{n-1}\frac{dy}{dx}\)

1

Core idea

When differentiating with respect to x, every function of y produces a factor dy/dx.

Use implicit differentiation to find gradients, tangents, normals and stationary points.

2

Key results and method

\[\frac{d}{dx}(y^n)=ny^{n-1}\frac{dy}{dx}\]
\[\frac{d}{dx}f(y)=f\'(y)\frac{dy}{dx}\]
  1. Differentiate both sides with respect to \(x\).
  2. Attach \(dy/dx\) to differentiated y-terms.
  3. Collect all \(dy/dx\) terms.
  4. Factor and solve for \(dy/dx\).
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Worked example

Question

For \(x^2+y^2=25\), find \(dy/dx\).

Solution
  1. \(2x+2y\,dy/dx=0\).
  2. Solve for \(dy/dx\).
Answer: \(\frac{dy}{dx}=-\frac{x}{y}\).
4

Practice

Try these questions before moving on.

  1. Find tangent and normal equations to the circle at \((3,4)\).
  2. Find stationary points on an implicitly defined curve.
  3. Differentiate a relation containing a product \(xy\).

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