Ex 5.6, 7 - Chapter 5 Class 12 Continuity and Differentiability
Last updated at April 16, 2024 by Teachoo
Derivatives in parametric form
Derivatives in parametric form
Last updated at April 16, 2024 by Teachoo
Ex 5.6, 7 If x and y are connected parametrically by the equations without eliminating the parameter, Find 𝑑𝑦/𝑑𝑥, 𝑥 =(〖𝑠𝑖𝑛〗^3 𝑡)/√(cos2𝑡 ) , 𝑦 = (〖𝑐𝑜𝑠〗^3 𝑡)/√(cos2𝑡 )Here, 𝑑𝑦/𝑑𝑥 = (𝑑𝑦/𝑑𝑡)/(𝑑𝑥/𝑑𝑡) Calculating 𝒅𝒚/𝒅𝒕 𝑦 = (〖𝑐𝑜𝑠〗^3 𝑡)/√(cos2𝑡 ) 𝑑𝑦/𝑑𝑡 " " = 𝑑/𝑑𝑡 ((〖𝑐𝑜𝑠〗^3 𝑡)/√(cos2𝑡 )) 𝑑𝑦/𝑑𝑡 " " = (𝑑(〖𝑐𝑜𝑠〗^3 𝑡)/𝑑𝑡 . √(cos2 𝑡) − 𝑑(√(cos2𝑡 ))/𝑑𝑡 .〖 cos^3〗𝑡)/(√(cos2 𝑡))^2 𝑑𝑦/𝑑𝑡 " = " (3 cos^2〖𝑡 〗. 𝑑(cos𝑡 )/𝑑𝑡. √(cos2 𝑡) − 1/(2√(cos2𝑡 )) . 𝑑(cos2𝑡 )/𝑑𝑡 .〖 cos^3〗𝑡)/(√(cos2 𝑡))^2 Using quotient rule As (𝑢/𝑣)^′ = (𝑢^′ 𝑣 − 𝑣^′ 𝑢)/𝑣^2 𝑑𝑦/𝑑𝑡 " = " (3 cos^2〖𝑡 〗. (−sin𝑡 ) . √(cos2 𝑡) − 1/(2√(cos2𝑡 )) . (−2 sin2𝑡) .〖 cos^3〗𝑡)/(√(cos2 𝑡))^2 𝑑𝑦/𝑑𝑡 " =" (−3 cos^2〖𝑡 〗 sin𝑡 √(cos2 𝑡) + 1/√(cos2𝑡 ) . sin2𝑡 .〖 cos^3〗𝑡)/(√(cos2 𝑡))^2 𝑑𝑦/𝑑𝑡 " =" ((−3 cos^2〖𝑡 〗 sin𝑡 √(cos2 𝑡) × √(cos2𝑡 ) + sin2𝑡 .〖 cos^3〗𝑡)/√(cos2𝑡 ))/(√(cos2 𝑡))^2 𝑑𝑦/𝑑𝑡 " =" (−3 cos^2〖𝑡 〗 sin𝑡 (cos2 𝑡) + sin2𝑡 .〖 cos^3〗𝑡)/((√(cos2 𝑡))^2 (√(cos2 𝑡)) ) 𝑑𝑦/𝑑𝑡 " =" ( cos^2𝑡 (−3 sin𝑡 .cos2𝑡 +〖 cos〗𝑡 . sin2𝑡 ))/((cos2 𝑡)^(3/2) ) Calculating 𝒅𝒙/𝒅𝒕 𝑥 = (〖𝑠𝑖𝑛〗^3 𝑡)/√(cos2𝑡 ) 𝑑𝑥/𝑑𝑡 = 𝑑/𝑑𝑥 ((〖𝑠𝑖𝑛〗^3 𝑡)/√(cos2𝑡 )) 𝑑𝑥/𝑑𝑡 = (𝑑(〖𝑠𝑖𝑛〗^3 𝑡)/𝑑𝑡 . √(cos2𝑡 ) − (𝑑(√(cos2𝑡 )) )/𝑑𝑥 . 〖 𝑠𝑖𝑛〗^3 𝑡 )/(√(cos2𝑡 ))^2 𝑑𝑥/𝑑𝑡 = (3 〖𝑠𝑖𝑛〗^2 𝑡 . (𝑑(sin𝑡 ) )/𝑑𝑡 . √(cos2𝑡 ) − 1/(2√(cos2𝑡 )) . (𝑑(cos2𝑡 ) )/𝑑𝑥 . 〖 𝑠𝑖𝑛〗^3 𝑡 )/(√(cos2𝑡 ))^2 𝑑𝑥/𝑑𝑡 = (3 〖𝑠𝑖𝑛〗^2 𝑡 . cos𝑡 . √(cos〖2 𝑡〗 ) − 1/(2√(cos〖2 𝑡〗 )) . (−sin2𝑡 ) . 2 . 〖 𝑠𝑖𝑛〗^3 𝑡 )/((cos〖2 𝑡〗 ) ) 𝑑𝑥/𝑑𝑡 = (3 〖𝑠𝑖𝑛〗^2 𝑡 . cos𝑡 . (√(cos2𝑡 )) . (√(cos2𝑡 )) + sin2𝑡 . 〖 𝑠𝑖𝑛〗^3 𝑡 )/((√(cos2𝑡 )) (cos2𝑡 ) ) 𝑑𝑥/𝑑𝑡 = (3 〖𝑠𝑖𝑛〗^2 𝑡 . cos𝑡 . cos2𝑡 + sin2𝑡 . 〖 𝑠𝑖𝑛〗^3 𝑡 )/(cos2𝑡 )^(3/2) 𝑑𝑥/𝑑𝑡 = (〖𝑠𝑖𝑛〗^2 𝑡 (3 cos𝑡 . cos2𝑡 + sin2𝑡 . sin𝑡 ) )/(cos2𝑡 )^(3/2) Finding 𝒅𝒚/𝒅𝒙 𝒅𝒚/𝒅𝒙 = ((cos^2𝑡 (−3 sin𝑡 .cos2𝑡 +〖 cos〗𝑡 . sin2𝑡 ))/((cos2 𝑡)^(3/2) ))/((〖𝑠𝑖𝑛〗^2 𝑡 (3 cos𝑡 . cos2𝑡 + sin2𝑡 . sin𝑡 ) )/(cos2𝑡 )^(3/2) ) 𝑑𝑦/𝑑𝑥 = (cos^2𝑡 (−3 sin𝑡 .cos2𝑡 +〖 cos〗𝑡 . sin2𝑡 ))/(〖𝑠𝑖𝑛〗^2 𝑡 (3 cos𝑡 . cos2𝑡 + sin2𝑡 . sin𝑡 ) ) 𝑑𝑦/𝑑𝑥 = (cos^2𝑡 (−3 sin𝑡 .cos2𝑡 +〖 cos〗𝑡 . sin2𝑡 ))/(〖𝑠𝑖𝑛〗^2 𝑡 (3 cos𝑡 . cos2𝑡 + sin2𝑡 . sin𝑡 ) ) 𝑑𝑦/𝑑𝑥 = cot^2 𝑡 ((−3 sin𝑡 .cos2𝑡 +〖 cos〗𝑡 . sin2𝑡)/(3 cos𝑡 . cos2𝑡 + sin2𝑡 . sin𝑡 )) Taking cos 2t common 𝑑𝑦/𝑑𝑥 = cot^2 𝑡 ((cos2𝑡 (−3 sin𝑡 + cos𝑡 sin2𝑡/cos2𝑡 ))/(cos2𝑡 (3 cos〖𝑡 〗+sin𝑡 . sin2𝑡/cos2𝑡 ) )) 𝑑𝑦/𝑑𝑥 = cot^2 𝑡 ((−3 sin𝑡 + cos𝑡 sin2𝑡/cos2𝑡 )/(3 cos〖𝑡 〗+〖 sin〗𝑡 . sin2𝑡/cos2𝑡 )) 𝑑𝑦/𝑑𝑥 = cot^2 𝑡 ((−3 sin𝑡 + cos𝑡 tan2𝑡)/(3 cos〖𝑡 〗+〖 sin〗𝑡 . tan2𝑡 )) Taking cos t common 𝑑𝑦/𝑑𝑥 = cot^2 𝑡 ((cos𝑡 (−3 sin𝑡/cos𝑡 + tan2𝑡))/(cos𝑡 (3 + sin𝑡/cos𝑡 . tan2𝑡 )) 𝑑𝑦/𝑑𝑥 = cot^2 𝑡 ((−3 tan𝑡 + tan2𝑡)/(3 +〖 tan〗𝑡 . tan2𝑡 )) 𝑑𝑦/𝑑𝑥 = cot^2 𝑡 ((tan2𝑡 − 3 tan𝑡)/(3 +〖 tan〗𝑡 . tan2𝑡 )) Using tan 2𝜃 = (2 𝑡𝑎𝑛𝜃)/(1 〖𝑡𝑎𝑛〗^2𝜃 ) 𝑑𝑦/𝑑𝑥 = cot^2 𝑡 (((2 tan𝑡)/(1 − tan^2𝑡 ) − 3 tan𝑡)/(3 + (tan𝑡 ) ((2 tan𝑡)/(1 −tan^2𝑡 )) )) 𝑑𝑦/𝑑𝑥 = cot^2 𝑡 (((2 tan𝑡 − 3 tan𝑡 (1 − tan^2𝑡 ))/((1 − tan^2𝑡)))/((3 (1− tan^2𝑡 ) + tan𝑡 (2 tan𝑡 ))/((1 − tan^2𝑡)))) 𝑑𝑦/𝑑𝑥 = cot^2 𝑡 ((2 tan𝑡 −3 tan𝑡 (1 − tan^2𝑡 ))/(3 (1− tan^2𝑡 ) + tan𝑡 (2 tan𝑡 ) )) 𝑑𝑦/𝑑𝑥 = cot^2 𝑡 ((2 tan𝑡 −3 tan𝑡 + 3 tan^3𝑡)/(3 − 3 tan^2𝑡 + 2 tan^2𝑡 )) 𝑑𝑦/𝑑𝑥 = cot^2 𝑡 ((−tan𝑡 + 3 tan^3𝑡 ) )/((3 −tan^2𝑡 ) ) 𝑑𝑦/𝑑𝑥 = cot^2 𝑡 (−(tan𝑡 −3 tan^3𝑡 ) )/((3 −tan^2𝑡 ) ) 𝑑𝑦/𝑑𝑥 = 〖−cot〗^2 𝑡 ((tan𝑡 −3 tan^3𝑡 ) )/((3 −tan^2𝑡 ) ) Multiplying cot2 t to numerator 𝑑𝑦/𝑑𝑥 = −((cot^2𝑡 × tan𝑡 − 3 cot^2𝑡 tan^3𝑡)/(3 −tan^2𝑡 )) 𝑑𝑦/𝑑𝑥 = − ((1/tan^2𝑡 × tan𝑡 − 3 × 1/tan^2𝑡 ×tan^3𝑡)/(3 −tan^2𝑡 )) 𝑑𝑦/𝑑𝑥 = − ((1/tan𝑡 .− 3 tan𝑡 )/(3 − tan^2𝑡 )) 𝑑𝑦/𝑑𝑥 = − (((1 −3 tan𝑡 (tan〖𝑡)〗)/tan𝑡 )/(3 − tan^2𝑡 )) 𝑑𝑦/𝑑𝑥 = − ((1 − 3 tan^2𝑡 )/(tan𝑡 (3 − tan^2𝑡 ) )) 𝑑𝑦/𝑑𝑥 = − ((1 − 3 tan^2𝑡 )/(3 tan𝑡 −tan^3𝑡 )) 𝑑𝑦/𝑑𝑥 = (−1)/(((3 tan𝑡 −〖 tan〗^3𝑡)/(1 −3 tan^2𝑡 )) ) 𝑑𝑦/𝑑𝑥 = − ((1 − 3 tan^2𝑡 )/(3 tan𝑡 −tan^3𝑡 )) 𝑑𝑦/𝑑𝑥 = (−1)/(((3 tan𝑡 −〖 tan〗^3𝑡)/(1 −3 tan^2𝑡 )) ) 𝐴𝑠 tan3𝑥=(3 tan𝑥 − tan^3𝑥)/(1 − 3 tan^2𝑥 ) 𝑑𝑦/𝑑𝑥 = (−1)/tan3𝑡 𝒅𝒚/𝒅𝒙 = −𝒄𝒐𝒕𝟑𝒕