Derivatives in parametric form

Chapter 5 Class 12 Continuity and Differentiability (Term 1)
Concept wise

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Misc 17 If π₯=π (cosβ‘π‘ + π‘ sinβ‘π‘) and y=π (sinβ‘π‘ β π‘ cosβ‘π‘), Find (π^2 π¦)/γππ₯γ^We need to find (π^2 π¦)/γππ₯γ^2 First we find ππ/ππ Here, ππ¦/ππ₯ = (ππ¦/ππ‘)/(ππ₯/ππ‘) Calculating ππ/ππ π¦=π (sinβ‘π‘β π‘ cosβ‘π‘ ) Differentiating π€.π.π‘. t ππ¦/ππ‘ = π(π (sinβ‘π‘β π‘ cosβ‘π‘ ))/ππ‘ ππ¦/ππ‘ = π π(sinβ‘π‘β π‘ cosβ‘π‘ )/ππ‘ ππ¦/ππ‘ = π (π(sinβ‘π‘ )/ππ‘ β π(π‘ cosβ‘π‘ )/ππ‘) ππ¦/ππ‘ = π (cosβ‘π‘β π(π‘ cosβ‘π‘ )/ππ‘) ππ¦/ππ‘ = π (cosβ‘π‘ β(ππ‘/ππ‘ . cosβ‘π‘+ (π cosβ‘π‘)/ππ‘ . π‘ )) Using Product rule As (π’π£)β = π’βπ£ + π£βπ’ ππ¦/ππ‘ = π (cosβ‘π‘ β(cosβ‘π‘+(γβsinγβ‘π‘ ) . π‘)) ππ¦/ππ‘ = π (cosβ‘π‘ β(cosβ‘π‘β(sinβ‘π‘ ) . π‘)) ππ¦/ππ‘ = π (cosβ‘π‘ βcosβ‘π‘+π‘ .sinβ‘π‘ ) ππ¦/ππ‘ = π (0+π‘ sinβ‘π‘ ) ππ/ππ = π .π.πππβ‘π Calculating ππ/ππ π₯=π (cosβ‘π‘+ π‘ sinβ‘π‘ ) Differentiating π€.π.π‘. t ππ₯/ππ‘ = π(π (cosβ‘π‘ + π‘ sinβ‘π‘)" " )/ππ‘ ππ₯/ππ‘ = π (π(cosβ‘π‘ + π‘ sinβ‘π‘)/ππ‘) ππ₯/ππ‘ = π (π(cosβ‘π‘)/ππ‘ + π(π‘ sinβ‘π‘)/ππ‘) ππ₯/ππ‘ = π (γβsinγβ‘π‘ + π(π‘ sinβ‘π‘ )/ππ‘) Using product rule As (π’π£)β = π’βπ£ + π£βπ’ ππ₯/ππ‘ = π (γβsinγβ‘π‘+(ππ‘/ππ‘ . sinβ‘π‘+ π(sinβ‘π‘ )/ππ‘ . π‘ )) ππ₯/ππ‘ = π (γβsinγβ‘π‘+(sinβ‘π‘+cosβ‘π‘ . π‘)) ππ₯/ππ‘= π (βsinβ‘π‘+sinβ‘π‘+π‘ .cππ β‘π‘ ) ππ/ππ = π .π.πππβ‘π Finding ππ/ππ ππ/ππ = (ππ/ππ)/(ππ/ππ) ππ¦/ππ₯ = (π" " .π‘.sinβ‘π‘)/(π" " .π‘.cosβ‘π‘ ) ππ/ππ = πππβ‘π Again Differentiating π€.π.π‘.π₯. π/ππ (ππ/ππ) = π(πππβ‘π)/ππ (π^2 π¦)/(ππ₯^2 ) = π(tanβ‘π‘)/ππ₯ (π^2 π¦)/(ππ₯^2 ) = π(tanβ‘π‘)/ππ₯ . ππ‘/ππ‘ (π^2 π¦)/(ππ₯^2 ) =sec^2β‘π‘ . ππ‘/ππ₯ (π^2 π¦)/(ππ₯^2 ) =sec^2β‘π‘ Γ· ππ/ππ (π^2 π¦)/(ππ₯^2 ) = sec^2β‘π‘ Γ· π.π.ππππ (π^2 π¦)/(ππ₯^2 ) = (sec^2β‘π‘ )/(π" " . π‘.cosβ‘π‘ ) "We have calculated" ππ/ππ " = " π" ".π‘.πππ β‘π‘ (π^2 π¦)/(ππ₯^2 ) = (sec^2β‘π‘ )/(π" " . π‘ Γ 1/secβ‘π‘ ) (π^π π)/(ππ^π ) = (γπππγ^πβ‘π )/(π" " . π) 2