The gradient function
- Differentiation finds the gradient function, written dy/dx or f′(x). It gives the gradient of the curve at any point.
- If y = axⁿ, then dy/dx = naxⁿ⁻¹: multiply by the power, then reduce the power by 1.
- The derivative of a constant is 0, and the derivative of kx is k.
- Differentiate each term separately: y = 5x² − 3x + 7 gives dy/dx = 10x − 3.
Negative and fractional powers
- Rewrite first. y = 4/x = 4x⁻¹, so dy/dx = −4x⁻² = −4/x².
- y = 6√x = 6x^(1/2), so dy/dx = 3x^(−1/2) = 3/√x.
- Expand brackets before differentiating: y = (x + 2)(x − 5) = x² − 3x − 10, so dy/dx = 2x − 3.
Using the gradient function
- To find the gradient at a point, substitute its x-value. For y = x³ − 2x at x = 2: dy/dx = 3x² − 2 = 10.
- To find where the gradient has a certain value, set dy/dx equal to it and solve. For y = x² + 5x − 2, a gradient of 9 means 2x + 5 = 9, so x = 2.
- A function is increasing where dy/dx > 0 and decreasing where dy/dx < 0.
Key terms
- Differentiation
- Finding the gradient function of a curve.
- dy/dx
- The gradient function of y with respect to x.
- Derivative
- The result of differentiating.
- Gradient of a curve
- The gradient of the tangent at a point.
- Increasing function
- A function whose gradient is positive.