Calculus
Differentiation Using The Chain Rule
Introduction To The Chain Rule - The Power Rule
y = (expression in x)n
dy/dx = n(expression in x)n-1(differential of the expression in x)
[1] Differentiate with respect to the brackets as a unit
[2] Multiply by the differential of the inside of the brackets
[3] Simplify
| EXAMPLE 1 | |
| Differentiate y = (2x)4 with respect to x | |
| [1] 'Differentiate with respect to the brackets' | dy/d(...) = 4(2x)3 |
| [2] Multiply by the 'inside differential' | dy/dx = 4(2x)3(2) |
| [3] Simplify | dy/dx = 4(2)(2x)3 |
| dy/dx = 8(2x)3 | |
A presentation that shows the method and works to avoid error is
y = (2x)4
dy/dx = 4(2x)3(2)
dy/dx = 4(2)(2x)3
dy/dx = 8(2x)3
| EXAMPLE 2 | |
| Differentiate y = 9(2x)4 with respect to x | |
| [1] 'Differentiate with respect to the brackets' | dy/d(...) = 4 × 9(2x)3 |
| [2] Multiply by the 'inside differential' | dy/dx = 4 × 9(2x)3(2) |
| [3] Simplify | dy/dx = 4(9)(2)(2x)3 |
| dy/dx = 72(2x)3 | |
A presentation that shows the method and works to avoid error is
y = 9(2x)4
dy/dx = 4(9)(2x)3(2)
dy/dx = 4(9)(2)(2x)3
dy/dx = 72(2x)3
| EXAMPLE 3 | |
| Differentiate y = (3x2)6 with respect to x | |
| [1] 'Differentiate with respect to the brackets' | dy/d(...) = 6(3x2)5 |
| [2] Multiply by the 'inside differential' | dy/dx = 6(3x2)5(2×3x) |
| [3] Simplify | dy/dx = 6(3x2)5(6x) |
| dy/dx = 6(6x)(3x2)5 | |
| dy/dx = 36x(3x2)5 | |
A presentation that shows the method and works to avoid error is
y = (3x2)6
dy/dx = 6(3x2)5(2×3x)
dy/dx = 6(3x2)5(6x)
dy/dx = 6(6x)(3x2)5
dy/dx = 36x(3x2)5
| EXAMPLE 4 | |
| Differentiate y = 5(2x3)7 with respect to x | |
| [1] 'Differentiate with respect to the brackets' | dy/d(...) = 7 × 5(2x3)6 |
| [2] Multiply by the 'inside differential' |
dy/dx = 7 × 5(2x3)6(3×2x2) |
| [3] Simplify | dy/dx = 35(2x3)6(6x2) |
| dy/dx = 35(6x2)(2x3)6 | |
| dy/dx = 210x2(2x3)6 | |
A presentation that shows the method and works to avoid error is
y = 5(2x3)7
dy/dx = 7 × 5(2x3)6(3×2x2)
dy/dx = 35(2x3)6(6x2)
dy/dx = 35(6x2)(2x3)6
dy/dx = 210x2(2x3)6
The Chain Rule As Shown Above May Be Called The Power Rule
This Video Shows 2 More Complicated Examples
The Next Video Shows 5 Power Rule Examples
A VERY STRONG SUGGESTION
- watch the first 2 questions
- with the next 3 questions
- run the video until you can see the question
- try that question on paper
- run the video to see if you are correct
Two More Difficult Examples
[1] 4 level Application Of The Chain Rule
[2] 4 level Application Of The Chain Rule With Change Of Base Of Log
With exponentials
With logarithms