Definition of Derivatives (Edexcel IAL Maths)

Spec point — 4.1

Definition of derivatives

What is the derivative of a function?

• Differentiation is an operation that calculates the rate of change of a function with respect to a variable

  • This means how much the function varies when the variable increases by one unit

• To differentiate a function (f(x)) with respect to the variable x we use the notation

  • $\frac{ \mathrm{d} }{ \mathrm{d} x} ( \mathrm{f} ( x ) )$

• The result is called the derivative

What is the link between derivatives and gradients?

• The rate of change of a function f(x) with respect to x can be thought of as the gradient function of the graph y = f(x)

  • We can write the gradient function (or derivative) as

    • $f'(x)$ or $\frac{dy}{dx}$

• The rate of change of a function (or the gradient of its graph) varies for different values of x

  • For a linear function f(x)* *= mx + c the gradient is constant

    • We can write this as $f'(x)=m$ or $\frac{dy}{dx}=m$

  • For the quadratic function f(x) = x² the gradient varies

    • Near the origin the gradient is close to 0
    • As x increases the gradient of the graph increases

How can I find the derivative of a function at a point?

• The derivative of a function (or gradient of its graph) at a point is equal to the gradient of the tangent to the graph at that point

• To estimate the gradient you could draw the tangent and calculate its gradient

• To find the actual gradient at a point x

  • Pick a second point on the curve close to the first point (call it x + h)

  • Calculate the gradient of the chord joining the two points

    • $\frac{ \mathrm{f} ( x + h ) - \mathrm{f} ( x )}{( x + h ) - x} = \frac{ \mathrm{f} ( x + h ) - \mathrm{f} ( x )}{h}$

  • Move the second point closer to the first point (make h get close to zero)

  • Examine what happens to the gradient of the chord

  • The gradient of the tangent will be the limit of the gradients of the chords

    • $\mathrm{f}^{'} ( x ) = \lim_{h \to 0} \frac{ \mathrm{f} ( x + h ) - \mathrm{f} ( x )}{h}$
    • You do not need to remember this formula

Worked Example

Exam Hint

• Deriving a derivative from scratch is not examinable
• This revision note is intended to give you an understanding of what derivatives do