Trigonometric Function Differentiation: Key Results and Techniques
Explore the essential teachings for exercising differentiation of trigonometric functions. Learn how to apply these results to various trigonometric functions and master differentiation using the chain rule and quotient rule. Enhance your understanding of differentiating sine, cosine, tangent, and cosecant functions in this comprehensive guide.
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Teachings for Exercise 9F Teachings for Exercise 9F
If:? = ?? If: ? =? ? ?? ??= ??? If:? = ????, then ?? ??+ ??? Differentiation ??? ?? ??? ?2 ?? ??= ?? ?? ??= ???2? ? = ???? ? = ???? You need to be able to apply to the results you have learnt to the remaining trigonometric functions Differentiate Differentiate ?? ??= ???? ?? ??= ???? Now we can replace all terms in the quotient rule Given that ? = ????, find ?? ??. ??? ?? ??? ?2 ?? ??= ?? We can rewrite ? = ???? as ? = quotient rule Sub in values ???? ????, and then use the ?? ??= ???? ???? (????)( ????) (????)2 Expand brackets on the numerator ?? ??=???2? + ???2? ???2? The top can be replaced since ???2? + ???2? 1 ?? ??= 1 ???2? Rewrite ?? ??= ???2? 9F
If:? = ?? If: ? =? ? ?? ??= ??? If:? = ????, then ?? ??+ ??? Differentiation ??= ? (?)???2?(?) ??? ?? ??? ?2 ?? ??= ?? ?? ??= ???2? If:? = ????(?), then ?? ? = ???? ? But what about if tan was of a function of x? Let ? = ?(?) ? = ?(?) ? = ???? Differentiate (we know this result from before) Differentiate ?? ??= ? (?) ?? ??= ???2? ?? ??=?? ?? ?? ?? Sub in values ?? ??= ???2?? (?) ? = ?(?) ?? ??= ???2?(?)? (?)
If:? = ?? If: ? =? If:? = ??????, then ?? ??= ?????????? ? ?? ??= ??? ??+ ??? Differentiation ??? ?? ??? ?2 ?? ??= ?? ?? 1 ? = You need to be able to apply to the results you have learnt to the remaining trigonometric functions ???? Write as a power ? = (????) 1 Differentiate using the chain rule ?? ??= (????) 2(????) Given that ? = ??????, find ?? ??. Rewrite as fractions ?? ??= 1 ???2? ???? Separate the sin terms We can rewrite ? = ?????? as ? = chain rule ?? ??= ???? ???? 1 1 ????, and then use the ???? Rewrite both ?? ??= ?????????? 9F
If:? = ?? If: ? =? If:? = ??????, then ?? If:? = ??????(?), then ?? ??= ?????????? ??= ? (?)??????(?)????(?) ? ?? ??= ??? ??+ ??? Differentiation ??? ?? ??? ?2 ?? ??= ?? ?? ? = ?????? ? But what about if cosec was of a function of x? Let ? = ?(?) ? = ?(?) ? = ?????? Differentiate (we know this result from before) Differentiate ?? ??= ? (?) ?? ??= ?????????? ?? ??=?? ?? ?? ?? Sub in values ?? ??= ??????????? (?) ? = ?(?) ?? ??= ??????(?)????(?)? (?)
If:? = ?? If: ? =? ? ?? ??= ??? If:? = ????, then ?? ??+ ??? Differentiation ??? ?? ??? ?2 ?? ??= ?? ?? ??= ???????? 1 ? = You need to be able to apply to the results you have learnt to the remaining trigonometric functions ???? Write as a power ? = (????) 1 Differentiate using the chain rule ?? ??= (????) 2( ????) Given that ? = ????, find ?? ??. Rewrite as fractions ?? ??= 1 ???2? ???? Separate the cos terms We can rewrite ? = ???? as ? = chain rule ?? ??= ???? ???? 1 1 ????, and then use the ???? Rewrite both ?? ??= ???????? 9F
If:? = ?? If: ? =? ? ?? ??= ??? If:? = ????, then ?? ??+ ??? Differentiation ??? ?? ??? ?2 ?? ??= ?? ?? ??= ???????? If:? = ????(?), then ?? ??= ? (?)????(?)????(?) ? = ???? ? But what about if sec was of a function of x? Let ? = ?(?) ? = ?(?) ? = ???? Differentiate (we know this result from before) Differentiate ?? ??= ? (?) ?? ??= ???????? ?? ??=?? ?? ?? ?? Sub in values ?? ??= ????????? (?) ? = ?(?) ?? ??= ???? ? ????(?)? (?)
If:? = ?? If: ? =? If:? = ????, then ?? ??= ?????2? ? ?? ??= ??? ??+ ??? Differentiation ??? ?? ??? ?2 ?? ??= ?? ?? 1 ? = You need to be able to apply to the results you have learnt to the remaining trigonometric functions ???? Write as a power ? = (????) 1 Differentiate using the chain rule ?? ??= (????) 2(???2?) Given that ? = ????, find ?? ??. Rewrite as fractions 2 ?? ??= ??= ???2? ???? ???? 1 ???2? Rewrite the first term We can rewrite ? = ???? as ? = chain rule ?? 1 1 ???2? ????, and then use the ???2? The ???2? terms will cancel out ?? ??= 1 ???2? Rewrite ?? ??= ?????2? 9F
If:? = ?? If: ? =? If:? = ????, then ?? If:? = ????(?), then ?? ??= ?????2? ??= ? (?)?????2?(?) ? ?? ??= ??? ??+ ??? Differentiation ??? ?? ??? ?2 ?? ??= ?? ?? ? = ???? ? But what about if cot was of a function of x? Let ? = ?(?) ? = ?(?) ? = ???? Differentiate (we know this result from before) Differentiate ?? ??= ? (?) ?? ??= ?????2? ?? ??=?? ?? ?? ?? Sub in values ?? ??= ?????2?? (?) ? = ?(?) ?? ??= ?????2?(?)? (?)
If:? = ?? If: ? =? ? ?? ??= ??? ??+ ??? Differentiation ??? ?? ??? ?2 ?? ??= ?? ?? You need to be able to apply to the results you have learnt to the remaining trigonometric functions You need to know the derivatives of sine and cosine, but the rest you are given in the formula booklet! 9F
If:? = ?? If: ? =? If:? = ????(?), then ?? ??= ? (?)???2?(?) ? ?? ??= ??? ??+ ??? Differentiation ??? ?? ??? ?2 ?? ??= ?? ?? ? = ? ? = ???2? You need to be able to apply to the results you have learnt to the remaining trigonometric functions Differentiate Differentiate ?? ??= 1 ? = 2???22? Now we can replace all terms in the product rule Differentiate: ?? ??= ??? ??+ ??? Sub in expressions from above ?? ? = ????2? ?? ??= ? 2???22? + (???2?)(1) You need to use the product rule here! Simplify ?? ??= 2????22? + ???2? 9F
If:? = ?? If: ? =? If:? = ????(?), then ?? ??= ? (?)???2?(?) ? ?? ??= ??? ??+ ??? Differentiation ??? ?? ??? ?2 ?? ??= ?? ?? ? = ???4? You need to be able to apply to the results you have learnt to the remaining trigonometric functions Write as a power ? = (????)4 Differentiate using the chain rule ?? ??= 4(????)3???2? Simplify Differentiate: ?? ??= 4???3????2? ? = ???4? You need to use the chain rule here! 9F
If:? = ?? If: ? =? If:? = ??????(?), then ?? ??= ? (?)??????(?)????(?) ? ?? ??= ??? ??+ ??? Differentiation ??? ?? ??? ?2 ?? ??= ?? ?? ? = ?2 ? = ?????2? You need to be able to apply to the results you have learnt to the remaining trigonometric functions Differentiate Differentiate ?? ??= 2?????2????2? ?? ??= 2? Now we can replace all terms in the quotient rule Differentiate: ??? ?? ??? ?2 ?? ??= ?? ? =?????2? Sub in values using the expressions above ?2 ?2 ?? ??= 2?????2????2? (?????2?)(2?) ?2 2 You need to use the quotient rule here! Expand bracket on the numerator ?? ??= 2?2?????2????2? 2??????2? ?4 Divide all by ? ?? ??= 2??????2????2? 2?????2? ?3 Factorise 2?????2? on the numerator ?? ??= 2?????2?(????2? + 1) ?3 9F
If:? = ?? If: ? =? If:? = ????, then ?? ??= ???????? ? ?? ??= ??? ??+ ??? Differentiation ??? ?? ??? ?2 ?? ??= ?? ?? You need to be able to apply to the results you have learnt to the remaining trigonometric functions ? = ???3? Rewrite ? = (????)3 Differentiate using the chain rule ?? ??= 3(????)2(????????) Differentiate: Simplify ?? ??= 3???3???? ? = ???3? You need to use the chain rule here! 9F
