Definition of Derivatives (Cambridge (CIE) AS Maths: Pure 1): Revision Note

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

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

• 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

    • $\mathrm{f}\text{'}\left(x\right)$ 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 $\mathrm{f}\text{'}\left(x\right)=m$ 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

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

  • 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

    • ${\mathrm{f}}^{\text{'}}\left(x\right)=\underset{h\to 0}{\lim }\frac{\mathrm{f}\left(x+h\right)-\mathrm{f}\left(x\right)}{h}$

    • You do not need to remember this formula