Differentiation

The value of a function changes when one of the variables (x, y, a, or b etc) changes. It may change like the variable, (both doubling) or it may change more, or less. Differentiation means finding the derivative of an expression.

This means you are finding a rate of change for a variable. It expresses the slope of the curve.

The derivative of velocity with respect to time is acceleration. If you map the profit of a company over time, the derivative will tell you if the company is getting stronger, levelling off or heading backwards.

• D1 Limit of a function

The limit of a function means finding the value of a curve at a particular point. But what is that point? A value to many decimal places can be very long, to define a value that is very precise. The precise point can be infinitely small.

We can only ask what the value of the curve is as we get closer to that point, from values above and from values below. That is what we mean by approaching the limit.

• D2 Gradients, tangents and derivatives

A tangent is a line that touches a curve at only one point. Where that point sits along the function curve, determines the slope (i.e. the gradient) of the tangent to that point. (See also Functions and graphs)

A derivative of a function gives you the gradient of a tangent at a certain point on a curve. If you plug the x value into the derivative function, you will get the slope of the tangent at that point, defined by the x value. Practice evaluating the gradients of these tangents to a curve.

• D3 Differentiation from first principles

Learn how to take a derivative of a function using first principles. Using this method is the best way to understand the concepts around differentiation. Start here to really appreciate what you are doing when you differentiate, before you start differentiating using other methods in later modules.

• D4 Rules for differentiation

Learn about the rules for differentiation and the different notations that are used. This section includes algebraic, exponential, logarithmic and trigonometric examples.

• D5 The chain rule

How do you differentiate a larger function that has components that are smaller functions? This module will show you how to package these functions and work them out separately, before plugging them into the Chain Rule formula.

• D6 The product rule

What is the product rule? It is useful when you want to differentiate a function that comprises one function multiplied by another function.

• D7 Quotient rule

What is the quotient rule? The quotient rule is like the product rule but this time it is for one function that is divided by another (rather than multiplied).
Review this section to learn how to differentiate using the quotient rule.

• D8 Maxima and minima

How do you find the maximum (highest) or minimum (lowest) value of a curve? The maximum or minimum values of a function occur where the derivative is zero. That is where the graph of the function has a horizontal tangent. If you go looking for the horizontal tangents (i.e. where the derivative = 0), you will be able to pinpoint the maxima or minima of a curve.

• D9 Curve sketching

If you find some key points of a function such as: maxima, minima, or turning points; x and y axis intercepts; and regions where the gradient is positive or negative, you can put together a sketch of a curve. Read this section to find examples of this being done.

• D10 Rates of change

If there is a relationship between two or more variables (like, area and radius of a circle (A = πr2 ), or pressure, volume and temperature of a gas), then there will also be a relationship between how these variables change. You may need to find how fast one variable changes in relation to another variable that is changing. This is called the rate of change.

• D11 Small changes and approximations

Sometimes a small change in one variable can render a big change in a larger value. For example, a small increase (or error) in the radius of a sphere means a lot more volume is added!
If you estimate the small error in one variable, you can calculate the significant change in a larger variable by using derivatives.

• D12 Implicit differentiation

What is implicit differentiation? How do you differentiate a function (say y=) that has two different variables in it (say x and k)? What do you do when you cannot express one as a function of the other? The technique of implicit differentiation is then useful.

Implicit differentiation enables you to find the derivative of y with respect to x without having to solve the original equation for y (see also Rearranging formula). Implicit differentiation may require using the Chain rule and Product Rule together.

Read this sheet for some worked examples and exercises in this process.

• D13 Partial differentiation

What is partial differentiation? You may have an expression with 3 variables or more. The volume (V) of a cylinder for example depends on radius(r.) and height(h).

It makes it easier to hold one value as constant and differentiate with respect to the other two. This helps us to see how the change in one impacts the change in the other, without being muddled by the third variable.

Partial derivatives reveal how a function with many variables changes when you adjust just one of the variables in the input.

• D14 Higher Order Derivatives

Higher derivatives are used in many mathematical, scientific and engineering subjects.

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