Differentiation

Differentiation from First Principles

Differentiation is about rates of change for example, the slope of a line is the rate of change of y with respect to x. To find the rate of change of a more general function, it is necessary to take a limit. This is done explicitly for a simple quadratic function.

Differentiating Powers of x

Differentiating powers of x. If y=x^n then dy/dx = nx^n-1. This result is derived in this video and illustrated with several examples, including cases where n is negative, or is a fraction.

Differentiating sines and cosines

Differentiating sines and cosines. This video covers the differentiation of sin x and cos x from first principles.

Differentiating logs and exponents

Differentiating logs and exponentials. This unit gives details of how logarithmic functions (ln x) and exponential functions (e^x) are differentiated.

Using a Table of Derivatives

Using a table of derivatives. This video looks at the construction of a Table of Derivatives of common functions using differentiation from first principles. Linearity rules for constant multiples of functions and for sum/difference of two functions, with examples, are explored. The table is extended using the chain rule for differentiation.

The Quotient rule

The quotient rule. This is a special rule that may be used to differentiate the quotient of two functions.

The Product Rule

The product rule. This is a special rule that may be used to differentiate the product of two (or more) functions.

The Chain Rule

The chain rule. This is a special rule that may be used to differentiate a composite function (that is, a function of another function).

Parametric Differentiation

Parametric differentiation. Instead of a function y(x) being defined explicitly in terms of the independent variable x, it is sometimes useful to define both x and y in terms of a third variable, t say, known as a parameter. Functions defined in this way can be differentiated using a technique called parametric differentiation.

Differentiation by Taking Logarithms

Differentiation by taking logarithms. The logarithm of a product is a sum, and so logarithms can be used to simplify certain functions for the purpose of differentiation.

Implicit Differentiation

Implicit differentiation . Sometimes functions are not given in the form y = f(x) but in a more complicated form where it is difficult to express y explicitly in terms of x. These are called implicit functions, and they can be differentiated to give dy/dx.

Extending the Table of Derivatives

Extending the table of derivatives. In this video the Table of Derivatives continues to be built using rules described in other units.

Tangents and Normals

Tangents and normals. This unit explains how differentiation can be used to calculate the equations of the tangent and normal to a curve. The tangent is a straight line which just touches the curve at a given point. The normal is a straight line through the point perpendicular to the tangent.

Maxima and minima

Maxima and minima. Because the derivative provides information about the gradient of a graph of a function, we can use it to locate points on a graph where the gradient is zero. Such points are often associated with the largest or smallest values of the function, at least in their immediate locality.