Study guides › Level 10 · Calculus

Slope of the touching line (tangent)

Level 10 · Calculus · for questions like “What is the slope of the tangent to y = x² + 3x at x = 2?”

💡 The idea

The slope of the curve at a point is the derivative at that point. The line that just touches the curve there (the tangent) has that slope.

🪜 The steps

1. Take the derivative.
2. Put the x in.
3. A flat point (slope 0): set the derivative equal to 0 and solve.

✏️ Worked examples

1. y = x² + 3x at x = 2: dy/dx = 2x + 3, so the slope is 4 + 3 = 7.

2. Where is y = x² − 8x + 1 flat? 2x − 8 = 0, so x = 4.

⚠️ Watch out

• Put the x into the derivative, not into the original curve. The curve's height is something else.

🎯 Try these

1. y = x² − 4x at x = 5: slope?
2. y = x³ at x = 2: slope?
3. Where is y = x² + 10x flat?

Answers: 1) 6 2) 12 3) x = −5

This guide is also in the 10MathQ app: tap “📖 Study guide” after a question.