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

Gradient

• 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?changes

   

• 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?gradient function?is an expression that allows the gradient to be calculated anywhere along a curve
• The?gradient function?is also called the derivative

How do I find the gradient function or derivative?

• This is really where the fun with differentiation begins!
• The?derivative?(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

• This method applies to positive and negative?integers
• Negative powers arise with fractions and?reciprocals

How do I find the value of a gradient?

• Substitute the?x?value into the?expression?for the?derivative, and?evaluate?it