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
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
How do I find the gradient of a curve using algebra?
This is really where the fun begins!
Drawing tangents each time you want the gradient of a curve is too much effort
It would be great if you could do it using algebra instead
The equation of a curve can be given in the form
Inputting x-coordinates gives outputs of y-coordinates
It is possible to create an algebraic function that take inputs of x-coordinates and gives
outputs of gradients
All of this is done without needing to sketch any graphs
This type of function has a few commonly used names:
The gradient function
The derivative
The derived function
The way to write this function is
This is pronounced "dy by dx"
In function notation, it can be written
pronounced f-dashed-of-x
To get from
to
you need to do an operation called differentiation
Differentiation turns curve equations into gradient functions
The main rule for differentiation is shown
This looks worse than it is!
For powers of x
STEP 1 Multiply the number in front by the power
STEP 2 Take one off the power (reduce the power by 1)
2x6 differentiates to 12x5
Note the following:
kx differentiates to k
so 10x differentiates to 10
any number on its own differentiates to zero
so 8 differentiates to 0
How do I use the gradient function to find gradients of curves?
Find the x-coordinate of the point on the curve you're interested in
Use differentiation to find the gradient (derived) function,
Substitute the x-coordinate into the gradient (derived) function to find the gradient
Exam Tip
When differentiating long awkward expressions, write each step out fully and simplify the
numbers after
Don't forget to write the left-hand sides of y = .... and
equation with the gradient function
= ... to avoid mixing up the curve
Worked example