# Differentiation inquiry

# The prompt

**Mathematical inquiry processes:**** **Explore, change representations; reason and prove. **Conceptual field of inquiry:** Differentiation; gradient function; graphs of quadratic and cubic functions.

The prompt follows on from the **quadratic function inquiry**. Once again it is not possible to satisfy the conditions in the prompt - that is, the gradient of the curve of a quadratic function cannot be the same for two values of x.

The inquiry introduces the derivative of a function with respect to *x*. In the context of the graph of *y* = *ax*^{2 }+ *bx *+ *c,* the derivative is the gradient function. As the notation is **arbitrary**, the teacher should inform students of its meaning in the orientation phase of the inquiry.

Gradient function

Describing *f'(x)* as the gradient function immediately suggests a line of inquiry. Students might independently start to wonder about the gradient at *x* = 1 and *x* = 2 or be guided to do so. The teacher should provide students with the graph of *y* = *x*^{2 }at the start of the inquiry (with an elongated *x*-axis).

The students might have already met the concept of a tangent to a curve, which they draw by eye and work out the gradient of the straight line. They soon realise that the gradient is different for each value of *x* and begin to conjecture a relationship between the value of *x* and the gradient at each point.

September 2022

# Lines of inquiry

1. Student-led exploration

Students use the **regulatory cards** to decide on the next stage of the inquiry. A common approach is to explore gradients of different parabolas. Students consider points on the graphs of other functions, such as *f(x)* = 2*x*^{2}, *f(x)* = 3*x*^{2} and *f(x)* = 4*x*^{2}.

They generalise from their results to derive a general formula for the derivative of functions in the form *f(x)* = *a**x*^{2}, which is *f'(x)* = 2*a**x. *The teacher chooses a pair of students to explain the generalisation so that the whole class has a model to work towards in further exploration.

The teacher coordinates suggestions for more lines of inquiry:

What happens if the function is in the form

*f(x)*=*a**x*^{2}^{ }+*c*? Why does the formula remain the same?What happens if the function is in the form

*f(x)*=*ax*^{2 }+*bx*? Why is the gradient function now*f('x)*= 2*a**x*+*b*? What is the connection between the change to the formula and the translation of the curve?

For example, the illustration below shows the parabolas of *f(x)* = *x*^{2}^{ }and *f(x)* = *x*^{2} + 2*x*. The second function can be re-written as *f(x)* = (*x* + 1)^{2} - 1 and represents a translation of one unit left and one down. The gradient of the second parabola at *x* = -2 is equal to the gradient of the first one at *x* = -1.

What happens to the gradient function for higher-degree polynomials?

2. Teacher-directed derivation

An alternative line of inquiry is to derive the gradient function by considering the chord *AB*. As *B* approaches *A*, the gradient of the chord approximates to the gradient of the tangent at *A. *

The teacher directs students to complete a table of results for different points on the graph of *f(x)* = *x*^{2}*. *When the coordinates of A are (1,1), the gradient is 2 at the limit (see below).

3. Finding functions for which *f*'(1) = *f*'(2)

Another line of inquiry involves an algebraic approach to find a cubic function for which *f*'(1) = *f*'(2). See the **mathematical notes**.