# More Advanced Differentiation

### The Product and Quotient Rules

### Video

This video from *Khan Academy* shows how to use to product rule to compute the derivative of a function that is a product of two simpler functions.

Note: All *Khan Academy* content is available for free at www.khanacademy.org.

This video from *PatrickJMT* gives more examples of computing derivatives with the product rule.

Note: All *patrickJMT* content is available for free at www.youtube.com/user/patrickJMT.

Patrick JMT’s Website: www.patrickjmt.com.

This video from *Khan Academy* gives an example of calculating a derivative using the quotient rule.

Note: All *Khan Academy* content is available for free at www.khanacademy.org.

### Notes

These notes from MathCentre introduce the product and quotient rules, along with some worked examples. There are exercises at the end for you to use for practice.

A pdf of this document can be downloaded here.

### Interactive Exercises

You can use these NUMBAS exercises to practise using the product and quotient rules to differentiate functions.

Please click here to view this NUMBAS question in another window.

Please click here to view this NUMBAS question in another window.

Please click here to view this NUMBAS question in another window.

Please click here to view this NUMBAS question in another window.

### The Chain Rule

### Video

This video from *Khan Academy* explains the concept behind the chain rule and goes through an example of computing a derivative using the rule.

Note: All *Khan Academy* content is available for free at www.khanacademy.org.

This video from *Krista King* gives an introductory example of differentiating a function using the chain rule.

Note: All *Krista King* content is available for free at www.youtube.com/c/Integralcalc.

Krista King’s Website: www.kristakingmath.com .

This video from *Khan Academy* discusses some common misconceptions regarding the chain rule and some common mistakes that are made when answering chain rule questions.

Note: All *Khan Academy* content is available for free at www.khanacademy.org.

This example video from *Khan Academy* answers in two different ways a differentiation problem that requires both the product rule and the chain rule.

Note: All *Khan Academy* content is available for free at www.khanacademy.org.

### Notes

These notes from MathCentre introduce the chain rule and the idea behind it. They also include some worked examples and exercises.

A pdf of this document can be downloaded here.

### Interactive Exercises

### Implicit Differentiation

### Videos

This video from *Khan Academy* discusses an example of implicit differentiation for the circle and relates the result to the geometry of the circle at the end.

Note: All *Khan Academy* content is available for free at www.khanacademy.org.

This video from *patrickJMT* discusses some more complicated examples of implicit differentiation.

Note: All *patrickJMT* content is available for free at www.youtube.com/user/patrickJMT.

Patrick JMT’s Website: www.patrickjmt.com.

### Notes

These notes from MathCentre introduce the concept of implicit differentiation and work through some examples. There are exercises with solutions that you can use for practice.

A pdf of this document can be downloaded here.

### Interactive Exercises

This NUMBAS exercise gives you an implicit polynomial relationship between $x$ and $y$ and asks you to compute the derivative in terms of both variables.

Please click here to view this NUMBAS question in another window.

### Parametric Differentiation

### Videos

These videos from *Ciaran O’Sullivan* introduces the idea of the parametric form for a function and its application in mechanics and engineering problems. The video finishes by introducing how to compute the slope of the tangent in terms of the given parameter.

This video from *Khan Academy* explains how to use parametric differentiation to compute the slope of the tangent line at a point when $x$ and $y$ are given in terms of a parameter $t$.

Note: All *Khan Academy* content is available for free at www.khanacademy.org.

### Notes

These notes from MathCentre introduce the idea of parametrising a curve and show how to differentiate a function defined parametrically. There is a short section at the end on second-order parametric derivatives, along with some exercises with answers given for you to use for practice.

A pdf of this document can be downloaded here.

### Interactive Exercises

These NUMBAS exercises ask you to compute the parametric derivative $\dfrac{dy}{dx}$ when $x$ and $y$ are expressed in terms of a parameter $t$.

Please click here to view this NUMBAS question in another window.

Please click here to view this NUMBAS question in another window.

Please click here to view this NUMBAS question in another window.

### Logarithmic Differentiation

### Videos

This video from *Krista King* shows how to use logarithmic differentiation to simplify the differentiation of complicated functions.

Note: All *Krista King* content is available for free at www.youtube.com/c/Integralcalc.

Krista King’s Website: www.kristakingmath.com .

### Notes

These notes from MathCentre describe why logarithmic differentiation is useful, how the rules of logarithms are applied, and go through some examples of increasing difficulty.

A pdf of this document can be downloaded here.