Introduction to Differentiation (Cambridge (CIE) IGCSE Additional Maths): Revision Note

Definition of gradient

What is the gradient of a curve?

• At a given point the gradient of a curve is defined as the gradient of the tangent to the curve at that point

• A tangent to a curve is a line that just touches the curve at one point but doesn't cut the curve at that point

• A tangent may cut the curve somewhere else on the curve

• It is only possible to draw one tangent to a curve at any given point

• Note that unlike the gradient of a straight line, the gradient of a curve is constantly changing

Definition of derivatives

What is a derivative?

• Calculus is about rates of change

  • the way a car’s position on a road changes is its speed (velocity)

  • the way the car’s speed changes is its acceleration

• The gradient (rate of change) of a (non-linear) function varies with $x$

• The derivative of a function is a function that relates the gradient to the value of $x$

  • For example, the derivative of  $y=x^2$  is  $2x$

    • This means that when $x=1$, the gradient of $y=x^2$  is  $2\left(1\right)=2$

    • And when $x=5$, the gradient of $y=x^2$  is  $2\left(5\right)=10$

• The derivative is also called the gradient function

Differentiating powers of x

What is differentiation?

• Differentiation is the process of finding an expression for the derivative (gradient function) from the equation of a curve

  • The equation of the curve is written $y=...$ and the gradient function is written $\frac{\mathrm{d}y}{\mathrm{d}x}=...$

How do I differentiate powers of x?

• Powers of $x$ are differentiated according to the following formula:

  • If $y=a{x}^n$ then $\frac{\mathrm{d}y}{\mathrm{d}x}=an{x}^{n-1}$

    • e.g.  If $y=4x^3$ then $\frac{\mathrm{d}y}{\mathrm{d}x}=4\times 3\times x^{3-1}=12x^2$

    • you "bring down the power" then "subtract one from the power"

• Don't forget these two special cases:

  • If $y=ax$ then$\frac{\mathrm{d}y}{\mathrm{d}x}=a$

    • e.g.  If $y=6x$ then $\frac{\mathrm{d}y}{\mathrm{d}x}=6$

  • If $y=a$ then$\frac{\mathrm{d}y}{\mathrm{d}x}=0$

    • e.g.  If $y=5$ then $\frac{\mathrm{d}y}{\mathrm{d}x}=0$

  • These allow you to differentiate linear terms in $x$ and constants

• Functions involving fractions with denominators in terms of $x$ will need to be rewritten as negative powers of $x$ first

  • e.g.  If $y=\frac{4}{x}$ then rewrite as $y=4x^{-1}$ and differentiate

How do I differentiate sums and differences of powers of x?

•  The formulae for differentiating powers of $x$ work for a sum or difference of powers of $x$

  • e.g.  If$y=5x^4+3x^{-2}+4$ then $\frac{dy}{dx}=5\times 4x^{4-1}+3\times \left(-2\right)x^{-2-1}+0$ $\frac{dy}{dx}=20x^3-6x^{-3}$

  • This is sometimes referred to differentiating 'term-by-term'

• Products and quotients (divisions) cannot be differentiated in this way so they need expanding/simplifying first

  • e.g.  If $y=(2x-3)(x^2-4)$ then expand to $y=2x^3-3x^2-8x+12$ which is a sum/difference of powers of $x$ and can then be differentiated

What can I do with derivatives (gradient functions)?

• The derivative can be used to find the gradient of a function at any point

  • The gradient of a function at a point is equal to the gradient of the tangent to the curve at that point

  • A question may refer to the gradient of the tangent