What is differentiation?
- Differentiation is part of the branch of mathematics called Calculus
- It is concerned with the rate at which changes takes place – so has lots of real‑world uses:
- The rate at which a car is moving – ie. its speed
- The rate at which a virus spreads amongst a population
- To begin to understand differentiation you’ll need to understand gradient (see Finding Gradients of Non-Linear Graphs)
- Gradient generally means steepness.
- For example, the gradient of a road up the side of a hill is important to lorry drivers
- On a graph the gradient refers to how steep a line or a curve is
- It is really a way of measuring how fast y changes as x changes
- This may be referred to as the rate at which y
- So gradient is a way of describing the rate at which change happens
Straight lines and curves
- For a straight line the gradient is always the same (constant)
- Recall y= mx + c, where m is the gradient (see Straight Lines – Finding Equations)
- For a curve the gradient changes as the value of x changes
- At any point on the curve, the gradient of the curve is equal to the gradient of the tangent at that point
- A tangent is a straight line that touches the curve at one point
- The derived function (aka gradient function) is an expression that allows the gradient to be calculated anywhere along a curve
- The derived function is also called the derivative
How do I find the derived function or derivative?
- This is really where the fun with differentiation begins!
- The derived function (dy/dx) is found by differentiating y
- This looks worse than it is!
- For powers of x …
STEP 1 Multiply by the power
STEP 2 Take one off the power
How do I find the value of a gradient?
- Substitute the x value into the expression for the derived function and evaluate it
When differentiating long, awkward expressions, write each step out fully and simplify afterwards.