Maths That Models Change
Differentiation & Rates
14 min · 2 worked examplesStudent plan
A derivative is a rate of change. Once you can read one, half of engineering stops being descriptive and starts being predictive.
After this lesson you can
- Say what a derivative means in units, not just symbols
- Differentiate a simple polynomial
- Explain why a capacitor passes fast signals using dV/dt
The gradient at a point
A straight line has one gradient everywhere. A curve does not, its steepness changes as you move along it, so "the gradient" only means anything at a specific point.
Differentiation finds that gradient. Written dy/dx, it is the rate at which y changes as x changes, at the instant you ask.
The rule you will use most is the power rule: the derivative of xⁿ is nxⁿ⁻¹. Multiply by the power, then knock the power down by one. So x³ differentiates to 3x², and 5x² differentiates to 10x. A constant differentiates to zero, because a constant does not change.
In the full lesson
- The gradient at a point
- What it means physically
- Maxima and minima
This lesson comes with the Student plan. The first module of Maths for Engineers is free.