L4 Derivatives of Complex Trigonometric Functions and Relations - Part 1
Completion requirements
Unit 4B
Trigonometry Part 2
Lesson 4: Derivatives of Complex Trigonometric Functions and Relations
Trigonometric functions may contain multiple expressions and operations. In the previous Lesson, examples requiring the use of the different derivative rules were provided. In this Lesson, the derivative rules will continue to be used in
conjunction with other differentiation and simplification techniques.
Find the derivative of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»h«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«msqrt»«mi»csc«/mi»«mn»2«/mn»«mi»x«/mi»«/msqrt»«/mrow»«/mstyle»«/math».
Rewrite the function as «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»h«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«msup»«mfenced
open=¨[¨ close=¨]¨»«mrow»«mi»csc«/mi»«mi mathvariant=¨normal¨»(«/mi»«mn»2«/mn»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mfenced»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«/msup»«/mrow»«/mstyle»«/math», and use the chain rule to find
the derivative.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»h«/mi»«mo mathvariant=¨italic¨»`«/mo»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«/mrow»«/mstyle»«/math»derivative of the outside function «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mo»§#8729;«/mo»«/mstyle»«/math» derivative of the inside function
Simplify the expression by rationalizing the denominator.
The derivative of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»h«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«msqrt»«mi»csc«/mi»«mn»2«/mn»«mi»x«/mi»«/msqrt»«/mrow»«/mstyle»«/math» is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»h«/mi»«mo mathvariant=¨italic¨»`«/mo»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«mo»§#8722;«/mo»«mi»cot«/mi»«mi mathvariant=¨normal¨»(«/mi»«mn»2«/mn»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«msqrt»«mi»csc«/mi»«mi mathvariant=¨normal¨»(«/mi»«mn»2«/mn»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/msqrt»«/mrow»«/mstyle»«/math».
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»h«/mi»«mo mathvariant=¨italic¨»`«/mo»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«/mrow»«/mstyle»«/math»derivative of the outside function «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mo»§#8729;«/mo»«/mstyle»«/math» derivative of the inside function
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»h«/mi»«mo mathvariant=¨italic¨»`«/mo»«mfenced»«mi»x«/mi»«/mfenced»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«msup»«mfenced
open=¨[¨ close=¨]¨»«mrow»«mi»csc«/mi»«mi mathvariant=¨normal¨»(«/mi»«mn»2«/mn»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mfenced»«mrow»«mo»§#8722;«/mo»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«/mrow»«/msup»«mo»§#8729;«/mo»«mfenced»«mrow»«mo»§#8722;«/mo»«mi»csc«/mi»«mi
mathvariant=¨normal¨»(«/mi»«mn»2«/mn»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mi»cot«/mi»«mi mathvariant=¨normal¨»(«/mi»«mn»2«/mn»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mfenced»«mo»§#8729;«/mo»«mn»2«/mn»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mo»§#8722;«/mo»«mn»2«/mn»«mi»csc«/mi»«mi
mathvariant=¨normal¨»(«/mi»«mn»2«/mn»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mi»cot«/mi»«mi mathvariant=¨normal¨»(«/mi»«mn»2«/mn»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«mrow»«mn»2«/mn»«msqrt»«mi»csc«/mi»«mi mathvariant=¨normal¨»(«/mi»«mn»2«/mn»«mi»x«/mi»«mi
mathvariant=¨normal¨»)«/mi»«/msqrt»«/mrow»«/mfrac»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
Simplify the expression by rationalizing the denominator.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨center left¨»«mtr»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mo»§#8722;«/mo»«menclose notation=¨updiagonalstrike¨»«mn»2«/mn»«/menclose»«mi»csc«/mi»«mi
mathvariant=¨normal¨»(«/mi»«mn»2«/mn»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mi»cot«/mi»«mi mathvariant=¨normal¨»(«/mi»«mn»2«/mn»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«mrow»«menclose notation=¨updiagonalstrike¨»«mn»2«/mn»«/menclose»«msqrt»«mi»csc«/mi»«mi
mathvariant=¨normal¨»(«/mi»«mn»2«/mn»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/msqrt»«/mrow»«/mfrac»«mo mathcolor=¨#FF0000¨»§#8729;«/mo»«mfrac mathcolor=¨#FF0000¨»«msqrt»«mi»csc«/mi»«mi mathvariant=¨normal¨»(«/mi»«mn»2«/mn»«mi»x«/mi»«mi
mathvariant=¨normal¨»)«/mi»«/msqrt»«msqrt»«mi»csc«/mi»«mi mathvariant=¨normal¨»(«/mi»«mn»2«/mn»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/msqrt»«/mfrac»«/mtd»«/mtr»«mtr»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mo»§#8722;«/mo»«menclose
notation=¨updiagonalstrike¨»«mi»csc«/mi»«mi mathvariant=¨normal¨»(«/mi»«mn»2«/mn»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/menclose»«mi»cot«/mi»«mi mathvariant=¨normal¨»(«/mi»«mn»2«/mn»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«msqrt»«mi»csc«/mi»«mi
mathvariant=¨normal¨»(«/mi»«mn»2«/mn»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/msqrt»«/mrow»«menclose notation=¨updiagonalstrike¨»«mi»csc«/mi»«mi mathvariant=¨normal¨»(«/mi»«mn»2«/mn»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/menclose»«/mfrac»«/mtd»«/mtr»«mtr»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»§#8722;«/mo»«mi»cot«/mi»«mi
mathvariant=¨normal¨»(«/mi»«mn»2«/mn»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«msqrt»«mi»csc«/mi»«mi mathvariant=¨normal¨»(«/mi»«mn»2«/mn»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/msqrt»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
The derivative of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»h«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«msqrt»«mi»csc«/mi»«mn»2«/mn»«mi»x«/mi»«/msqrt»«/mrow»«/mstyle»«/math» is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»h«/mi»«mo mathvariant=¨italic¨»`«/mo»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«mo»§#8722;«/mo»«mi»cot«/mi»«mi mathvariant=¨normal¨»(«/mi»«mn»2«/mn»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«msqrt»«mi»csc«/mi»«mi mathvariant=¨normal¨»(«/mi»«mn»2«/mn»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/msqrt»«/mrow»«/mstyle»«/math».
Find the derivative of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»h«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mfrac»«mrow»«msup»«mi»sin«/mi»«mn»2«/mn»«/msup»«mi»x«/mi»«/mrow»«mrow»«mi
mathvariant=¨normal¨»(«/mi»«mn»1«/mn»«mo»§#8722;«/mo»«mi»cos«/mi»«mi»x«/mi»«msup»«mi mathvariant=¨normal¨»)«/mi»«mn»2«/mn»«/msup»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math».
To find the derivative, the quotient rule can be applied. However, the result is a very complex expression. Instead, simplify the function first, and then find the derivative.
A Pythagorean identity can be used, along with factoring, to make differentiation more manageable.
Apply the quotient rule to find the derivative of this simplified function.
The derivative of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»h«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mfrac»«mrow»«msup»«mi»sin«/mi»«mn»2«/mn»«/msup»«mi»x«/mi»«/mrow»«mrow»«mi mathvariant=¨normal¨»(«/mi»«mn»1«/mn»«mo»§#8722;«/mo»«mi»cos«/mi»«mi»x«/mi»«msup»«mi mathvariant=¨normal¨»)«/mi»«mn»2«/mn»«/msup»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math» is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»h«/mi»«mo mathvariant=¨italic¨»`«/mo»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«mfrac»«mrow»«mo»§#8722;«/mo»«mn»2«/mn»«mi»sin«/mi»«mi»x«/mi»«/mrow»«mrow»«mi mathvariant=¨normal¨»(«/mi»«mn»1«/mn»«mo»§#8722;«/mo»«mi»cos«/mi»«mi»x«/mi»«msup»«mi mathvariant=¨normal¨»)«/mi»«mn»2«/mn»«/msup»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math».
A Pythagorean identity can be used, along with factoring, to make differentiation more manageable.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»h«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«msup»«mi»sin«/mi»«mn»2«/mn»«/msup»«mi»x«/mi»«/mrow»«mrow»«mi
mathvariant=¨normal¨»(«/mi»«mn»1«/mn»«mo»§#8722;«/mo»«mi»cos«/mi»«mi»x«/mi»«msup»«mi mathvariant=¨normal¨»)«/mi»«mn»2«/mn»«/msup»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mn»1«/mn»«mo»§#8722;«/mo»«msup»«mi»cos«/mi»«mn»2«/mn»«/msup»«mi»x«/mi»«/mrow»«mrow»«mi
mathvariant=¨normal¨»(«/mi»«mn»1«/mn»«mo»§#8722;«/mo»«mi»cos«/mi»«mi»x«/mi»«msup»«mi mathvariant=¨normal¨»)«/mi»«mn»2«/mn»«/msup»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mi mathvariant=¨normal¨»(«/mi»«mn»1«/mn»«mo»+«/mo»«mi»cos«/mi»«mi»x«/mi»«mi
mathvariant=¨normal¨»)«/mi»«menclose notation=¨updiagonalstrike¨»«mi mathvariant=¨normal¨»(«/mi»«mn»1«/mn»«mo»§#8722;«/mo»«mi»cos«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/menclose»«/mrow»«mrow»«mi mathvariant=¨normal¨»(«/mi»«mn»1«/mn»«mo»§#8722;«/mo»«mi»cos«/mi»«mi»x«/mi»«mi
mathvariant=¨normal¨»)«/mi»«menclose notation=¨updiagonalstrike¨»«mi mathvariant=¨normal¨»(«/mi»«mn»1«/mn»«mo»§#8722;«/mo»«mi»cos«/mi»«mi»x«/mi»«/menclose»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mn»1«/mn»«mo»+«/mo»«mi»cos«/mi»«mi»x«/mi»«/mrow»«mrow»«mn»1«/mn»«mo»§#8722;«/mo»«mi»cos«/mi»«mi»x«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
Apply the quotient rule to find the derivative of this simplified function.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»h«/mi»«mo mathvariant=¨italic¨»`«/mo»«mfenced»«mi»x«/mi»«/mfenced»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mi»g«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mi»f«/mi»«mo
mathvariant=¨italic¨»`«/mo»«mfenced»«mi»x«/mi»«/mfenced»«mo»§#8722;«/mo»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mi»g«/mi»«mo mathvariant=¨italic¨»`«/mo»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«msup»«mfenced open=¨[¨ close=¨]¨»«mrow»«mi»g«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mi
mathvariant=¨normal¨»(«/mi»«mn»1«/mn»«mo»§#8722;«/mo»«mi»cos«/mi»«mi»x«/mi»«mi»)(«/mi»«mn»0«/mn»«mo»§#8722;«/mo»«mi»sin«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»§#8722;«/mo»«mi mathvariant=¨normal¨»(«/mi»«mn»1«/mn»«mo»+«/mo»«mi»cos«/mi»«mi»x«/mi»«mi»)(«/mi»«mn»0«/mn»«mo»+«/mo»«mi»sin«/mi»«mi»x«/mi»«mi
mathvariant=¨normal¨»)«/mi»«/mrow»«mrow»«mi mathvariant=¨normal¨»(«/mi»«mn»1«/mn»«mo»§#8722;«/mo»«mi»cos«/mi»«mi»x«/mi»«msup»«mi mathvariant=¨normal¨»)«/mi»«mn»2«/mn»«/msup»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mo»§#8722;«/mo»«mi»sin«/mi»«mi»x«/mi»«mo»+«/mo»«mi»sin«/mi»«mi»x«/mi»«mi»cos«/mi»«mi»x«/mi»«mo»§#8722;«/mo»«mi»sin«/mi»«mi»x«/mi»«mo»§#8722;«/mo»«mi»sin«/mi»«mi»x«/mi»«mi»cos«/mi»«mi»x«/mi»«/mrow»«mrow»«mi
mathvariant=¨normal¨»(«/mi»«mn»1«/mn»«mo»§#8722;«/mo»«mi»cos«/mi»«mi»x«/mi»«msup»«mi mathvariant=¨normal¨»)«/mi»«mn»2«/mn»«/msup»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mo»§#8722;«/mo»«mn»2«/mn»«mi»sin«/mi»«mi»x«/mi»«/mrow»«mrow»«mi
mathvariant=¨normal¨»(«/mi»«mn»1«/mn»«mo»§#8722;«/mo»«mi»cos«/mi»«mi»x«/mi»«msup»«mi mathvariant=¨normal¨»)«/mi»«mn»2«/mn»«/msup»«/mrow»«/mfrac»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
The derivative of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»h«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mfrac»«mrow»«msup»«mi»sin«/mi»«mn»2«/mn»«/msup»«mi»x«/mi»«/mrow»«mrow»«mi mathvariant=¨normal¨»(«/mi»«mn»1«/mn»«mo»§#8722;«/mo»«mi»cos«/mi»«mi»x«/mi»«msup»«mi mathvariant=¨normal¨»)«/mi»«mn»2«/mn»«/msup»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math» is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»h«/mi»«mo mathvariant=¨italic¨»`«/mo»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«mfrac»«mrow»«mo»§#8722;«/mo»«mn»2«/mn»«mi»sin«/mi»«mi»x«/mi»«/mrow»«mrow»«mi mathvariant=¨normal¨»(«/mi»«mn»1«/mn»«mo»§#8722;«/mo»«mi»cos«/mi»«mi»x«/mi»«msup»«mi mathvariant=¨normal¨»)«/mi»«mn»2«/mn»«/msup»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math».
Differentiate «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«mn»6«/mn»«mi mathvariant=¨normal¨»(«/mi»«mi»sin«/mi»«mi»x«/mi»«mo»+«/mo»«mi»cos«/mi»«mi»x«/mi»«msup»«mi
mathvariant=¨normal¨»)«/mi»«mn»4«/mn»«/msup»«/mrow»«/mstyle»«/math», and then simplify.
Use the chain rule to find the derivative.
This expression can be simplified further by expanding the expression in the first bracket. Once partially expanded, Pythagorean and double angle identities can be used.
The derivative of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«mn»6«/mn»«mi mathvariant=¨normal¨»(«/mi»«mi»sin«/mi»«mi»x«/mi»«mo»+«/mo»«mi»cos«/mi»«mi»x«/mi»«msup»«mi mathvariant=¨normal¨»)«/mi»«mn»4«/mn»«/msup»«/mrow»«/mstyle»«/math» is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mfrac»«mrow»«mi»d«/mi»«mi»y«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«/mfrac»«mo»=«/mo»«mn»24«/mn»«mi mathvariant=¨normal¨»(«/mi»«mn»1«/mn»«mo»+«/mo»«mi»sin«/mi»«mn»2«/mn»«mi»x«/mi»«mi»)(«/mi»«mi»cos«/mi»«mn»2«/mn»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math».
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mfrac»«mrow»«mi»d«/mi»«mi»y«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«/mfrac»«mo»=«/mo»«mn»24«/mn»«mi mathvariant=¨normal¨»(«/mi»«mi»sin«/mi»«mi»x«/mi»«mo»+«/mo»«mi»cos«/mi»«mi»x«/mi»«msup»«mi
mathvariant=¨normal¨»)«/mi»«mn»3«/mn»«/msup»«mo»§#8729;«/mo»«mfenced»«mrow»«mi»cos«/mi»«mi»x«/mi»«mo»§#8722;«/mo»«mi»sin«/mi»«mi»x«/mi»«/mrow»«/mfenced»«/mrow»«/mstyle»«/math»
This expression can be simplified further by expanding the expression in the first bracket. Once partially expanded, Pythagorean and double angle identities can be used.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mfrac»«mrow»«mi»d«/mi»«mi»y«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«/mfrac»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»24«/mn»«mi
mathvariant=¨normal¨»(«/mi»«mi»sin«/mi»«mi»x«/mi»«mo»+«/mo»«mi»cos«/mi»«mi»x«/mi»«msup»«mi mathvariant=¨normal¨»)«/mi»«mn»2«/mn»«/msup»«mfenced»«mrow»«mi»sin«/mi»«mi»x«/mi»«mo»+«/mo»«mi»cos«/mi»«mi»x«/mi»«/mrow»«/mfenced»«mo»§#8729;«/mo»«mfenced»«mrow»«mi»cos«/mi»«mi»x«/mi»«mo»§#8722;«/mo»«mi»sin«/mi»«mi»x«/mi»«/mrow»«/mfenced»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»24«/mn»«mi
mathvariant=¨normal¨»(«/mi»«msup»«mi»sin«/mi»«mn»2«/mn»«/msup»«mi»x«/mi»«mo»+«/mo»«mn»2«/mn»«mi»sin«/mi»«mi»x«/mi»«mi»cos«/mi»«mi»x«/mi»«mo»+«/mo»«msup»«mi»cos«/mi»«mn»2«/mn»«/msup»«mi»x«/mi»«mi»)(«/mi»«mi»sin«/mi»«mi»x«/mi»«mo»+«/mo»«mi»cos«/mi»«mi»x«/mi»«mi»)(«/mi»«mi»cos«/mi»«mi»x«/mi»«mo»§#8722;«/mo»«mi»sin«/mi»«mi»x«/mi»«mi
mathvariant=¨normal¨»)«/mi»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»24«/mn»«mi mathvariant=¨normal¨»(«/mi»«mn»1«/mn»«mo»+«/mo»«mn»2«/mn»«mi»sin«/mi»«mi»x«/mi»«mi»cos«/mi»«mi»x«/mi»«mi»)(«/mi»«msup»«mi»cos«/mi»«mn»2«/mn»«/msup»«mi»x«/mi»«mo»§#8722;«/mo»«msup»«mi»sin«/mi»«mn»2«/mn»«/msup»«mi»x«/mi»«mi
mathvariant=¨normal¨»)«/mi»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»24«/mn»«mi mathvariant=¨normal¨»(«/mi»«mn»1«/mn»«mo»+«/mo»«mi»sin«/mi»«mn»2«/mn»«mi»x«/mi»«mi»)(«/mi»«mi»cos«/mi»«mn»2«/mn»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
The derivative of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«mn»6«/mn»«mi mathvariant=¨normal¨»(«/mi»«mi»sin«/mi»«mi»x«/mi»«mo»+«/mo»«mi»cos«/mi»«mi»x«/mi»«msup»«mi mathvariant=¨normal¨»)«/mi»«mn»4«/mn»«/msup»«/mrow»«/mstyle»«/math» is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mfrac»«mrow»«mi»d«/mi»«mi»y«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«/mfrac»«mo»=«/mo»«mn»24«/mn»«mi mathvariant=¨normal¨»(«/mi»«mn»1«/mn»«mo»+«/mo»«mi»sin«/mi»«mn»2«/mn»«mi»x«/mi»«mi»)(«/mi»«mi»cos«/mi»«mn»2«/mn»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math».
Watch the video Chain Rule Examples for additional examples on using the chain rule to find the derivative of trigonometric functions.