








Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class
Last updated at May 29, 2023 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