Ex 5.2, 7

Transcript

Ex 5.2, 7 (Introduction)Differentiate the functions with respect to 𝑥  2√(cot ( 𝑥2 )) Finding derivative of cot x Let 𝑦=cot⁡𝑥 𝑦= (𝑐𝑜𝑠 𝑥)/(𝑠𝑖𝑛 𝑥) Let 𝑢 = cos⁡𝑥 & 𝑣=sin 𝑥 𝑦′ = (𝑢/𝑣)^′ = (𝑢^′ 𝑣 − 〖 𝑣〗^′ 𝑢)/𝑣^2 Now, 𝑢 = cos⁡𝑥 & 𝑣=sin 𝑥 𝑢′ = −sin 𝑥 & 𝑣′=cos⁡𝑥 Thus, 𝑦 = (𝑢^′ 𝑣 − 〖 𝑣〗^′ 𝑢)/𝑣^2 = (−sin 𝑥" " sin 𝑥" " − cos⁡𝑥 " " cos⁡𝑥)/(sin^2 𝑥) = (−(sin^2 𝑥" +" cos^2⁡𝑥))/(sin^2 𝑥) = (−1)/(sin^2 𝑥) = −cosec^2⁡𝑥 Thus, 𝑦′ = (𝑢^′ 𝑣 − 〖 𝑣〗^′ 𝑢)/𝑣^2 = (−sin 𝑥" " sin 𝑥" " − cos⁡𝑥 " " cos⁡𝑥)/(sin^2 𝑥) = (−(sin^2 𝑥" +" cos^2⁡𝑥))/(sin^2 𝑥) = (−1)/(sin^2 𝑥) = −cosec^2⁡𝑥 Ex 5.2, 7Differentiate the functions with respect to 𝑥  2√(cot ( 𝑥2 ))