# Differentiation

**AS Level**

- Understand and use

- the derivative of f(x) as the gradient of the tangent to the graph of y = f(x) at a general point (x, y);
- the gradient of the tangent as a limit;
- interpretation as a rate of change;
- sketching the gradient function for a given curve;
- second derivatives;
- differentiation from first principles for small positive integer powers of x

- Understand and use the second derivative as the rate of change of gradient
- Differentiate x
^{n}, for rational values of n, and related constant multiples, sums and differences - Apply differentiation to find gradients, tangents and normals, maxima and minima and stationary points
- Identify where functions are increasing or decreasing

**A Level**

- connection to convex and concave sections of curves and points of inflection
- Differentiate ekx and akx, sinkx, coskx ,tankx and related sums, differences and constant multiples
- Understand and use the derivative of lnx
- points of inflection
- Differentiate using the product rule, the quotient rule and the chain rule, including problems involving connected rates of change and inverse functions
- Differentiate simple functions and relations defined implicitly or parametrically, for first derivative only
- Construct simple differential equations in pure mathematics and in context, (contexts may include kinematics, population growth and modelling the relationship between price and demand)

### Differentiation

This RISP activity introduces the subject of differentiation. Rather than start from first principles or learning a rule, the activity suggests using a graphing package to generate data. Starting with a quadratic graph, students find the gradient of the curve using a straight line graph and are encouraged arrive at a rule for differentiating a power of x through pattern spotting.

The RISP resource, Differentiation 2, contains four further investigations to explore differentiation.

### Mostly Calculus Materials

The DfE Standards Unit: Improving Learning in Mathematics collection contains a range of activities to support the learning of calculus, including:

C3 Matching Functions and Derivatives allows students to practice differentiating quadratic functions and finding the values of a function and its derivative at specific points.

C4 Differentiating and Integrating Fractional and Negative Powers asks students to convert functions into an appropriate form for differentiating or integrating and to differentiate and integrate negative and fractional powers of x.

C5 Finding Stationary Points of Cubic Functions invites students to find the stationary points of a cubic functions and to determine the nature of the stationary points.

Additionally, the 'Mostly Algebra' collection contains the resource A14 Exploring Equations in Parametric Form, in which students find and determine the nature of stationary points when a function is given in parametric form and find the intercepts of the function.

### Differentiation

These fourteen resources from Mathcentre cover a range of topics including the chain rule, product rule and quotient rule, differentiating a range of functions and differentiation from first principles. Each resource contains detailed notes, examples and questions to be completed, with answers.

### Differentiation and Integration

This Active A level resource from Susan Wall contains eleven problems that require students to explore where turning points occur, match statements about functions, derivative functions and gradients, explore the tangent and normal to a curve, suggest a possible graph given information about the function, the gradient of the function and the rate of change of the gradient of the function.

In addition the Further Indices resource contains a large matching activity involving equivalent functions, differentials and integrals with a range of different indices.

### Differentiation

This interactive excel file from The Virtual Textbook plots both quadratic and cubic functions along with the graphs of their first and second differential equations. By specifying the values of the functions students can see the effect this has on the graphs of the function, first differential and the second differential. The resource also include a number of student worksheets.

### Finding the Minimum Value and Sketching a Quadratic Function

These two videos from Casio explore methods for finding the minimum value of a quadratic function, including how to draw the graph of the function on a graphical calculator to find the solution.

### Investigating Stationary Points and Finding the Value of the First Derivative at a Particular Point on the Curve

The first of these three videos from Casio explores the function y = x/(16+x^{2}), finding the stationary points by rewriting the equation and using the product rule. The video also shows how to find the value of the differential of y= (1+e^{3x})^{5/3} using the chain rule.

The two further videos explore graphical solutions when finding stationary points by using a calculator to draw the graph and find the coordinates of the maximum and minimum values, and how to use a graphic calculator to verify solutions.

### Find the Value of the First and Second Derivative of a Function in x, at Specified Values of x

These two instructional videos from Casio explain how to use a graphical calculator to find the value of the first and second derivatives of a given function at specific values for x, and how to find the derivatives at specific values of x in order to verify the solution.