Differentiation | Mathematics for GCSE/IGCSE - Year 11 PDF Download

What is Differentiation?

• Differentiation is a fundamental concept in Calculus, focusing on the rate of change in quantities.
• It finds extensive applications in real-world scenarios such as determining the speed of a car or the spread of a virus in a population.

• To begin to grasp differentiation, it's essential to comprehend gradients.

How are gradients related to rates of change?

• The term "gradient" typically denotes the steepness of a slope. 
  • For instance, lorry drivers consider the gradient of a road up a hill to gauge its steepness and plan their routes accordingly.

• On a graph, the gradient indicates the steepness of either a line or a curve.
  • Essentially, it measures how quickly the y-value changes in relation to changes in the x-value.
  • This measurement can be seen as the rate at which y changes.
  • Therefore, the gradient describes the rate at which change occurs.

How do I find the gradient of a curve using its graph?

• For a straight line the gradient is always the same (constant)
  • Recall y = mx + c, where m is the gradient

• As the value of x changes, the gradient of a curve also changes.
  • At any given point on the curve, the gradient of the curve matches the gradient of the tangent line at that point.
  • A tangent line is a straight line that intersects the curve at a single point.

How do I find the gradient of a curve using algebra? 

• Drawing tangents each time you need the gradient of a curve can be laborious.
• It would be convenient if you could obtain it using algebra instead.
• The equation of a curve can be expressed as y = f(x), where inputting x-coordinates yields y-coordinates.
• It's possible to create an algebraic function that takes x-coordinates as inputs and outputs gradients without graph sketches.
• Such a function is commonly known as the gradient function, the derivative, or the derived function.
• It's written as dx/dy, pronounced "dy by dx", or in function notation as f′(x), pronounced "f-dashed-of-x".
• To transition from  you need to perform an operation called differentiation, which converts curve equations into gradient functions.
• The primary rule for differentiation is presented.
• For powers of x: 
  • STEP 1: Multiply the coefficient by the power. 
  • STEP 2: Reduce the power by 1. 
• For instance, 2x⁶ differentiates to 12x⁵. 
  • Some notes to remember: Any constant multiplied by x differentiates to the constant itself, so 10x differentiates to 10. 
  • Any standalone constant differentiates to zero, so 8 differentiates to 0.

How do I use the gradient function to find gradients of curves?

To find the x-coordinate of the point on the curve you're interested in:

• First, determine the x-coordinate of the specific point.
• Then, use differentiation to find the gradient function, represented as dx/dy.
• Substitute the x-coordinate into the gradient function to find the gradient at that point.