The second derivative implicitly

To find d²y/dx² differentiate the expression for y′ again with respect to x, using the quotient rule, and substitute the value of y′ obtained before.

Worked example

For x² + y² = 25, y′ = −x/y. Then

• y″ = −[y − x·y′]/y² = −[y − x(−x/y)]/y² = −(x² + y²)/y³ = −25/y³ (using x² + y² = 25).
• At (3, 4): y′ = −3/4 and y″ = −25/64 < 0, so the curve is concave down there — consistent with the upper half of a circle.

Horizontal and vertical tangents

• Horizontal tangent where dy/dx = 0: the numerator vanishes (and the denominator does not).
• Vertical tangent where dy/dx is undefined: the denominator vanishes (and the numerator does not).
• On x² + y² = 25 the numerator x = 0 gives (0, ±5) with horizontal tangents, and the denominator y = 0 gives (±5, 0) with vertical tangents.