Summary
Differentiation involves finding the rate of change of one quantity with respect to another and is used to find gradients and turning points of functions.
- Derivative — the result of differentiating a function. Example: For y = x^2, the derivative dy/dx = 2x.
- Gradient — the slope of a curve at a particular point. Example: Substitute x into the differentiated equation to find the gradient.
- Turning Point — a point where the curve changes direction, either a maximum or minimum. Example: At turning points, dy/dx = 0.
- Tangent — a straight line that touches a curve at a point without crossing it. Example: The equation of a tangent can be found using differentiation.
Exam Tips
Key Definitions to Remember
- Derivative: The result of differentiating a function.
- Gradient: The slope of a curve at a specific point.
- Turning Point: A point where the curve changes direction.
- Tangent: A line that touches a curve at one point.
Common Confusions
- Confusing the derivative with the original function.
- Misidentifying maximum and minimum points.
Typical Exam Questions
- Find dy/dx for y = 3x^2 + 2x + 1? Answer: dy/dx = 6x + 2
- Find the coordinates where the curve y = 2x^2 - 9x + 12 is parallel to the x-axis? Answer: Set dy/dx = 0 and solve for x.
- What is the greatest height of a ball given h = 7t - 5t^2? Answer: Find dh/dt and set it to zero to find t, then substitute back to find h.
What Examiners Usually Test
- Ability to differentiate basic functions.
- Identifying and calculating turning points.
- Finding the equation of a tangent to a curve.