Examples

Example 1 (i)

Example 1 (ii)

Example 1 (iii)

Example 2 (i)

Example 2 (ii) Important

Example 2 (iii) Important

Example 2 (iv)

Example 2 (v)

Example 3 (i) Important

Example 3 (ii) Important

Example 4 (i)

Example 4 (ii) Important

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Example 18

Example 19 (i) Important

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Example 20 (ii) Important

Example 21 (i)

Example 21 (ii) Important

Example 22 (i) You are here

Example 22 (ii) Important

Last updated at April 16, 2024 by Teachoo

Example 22 Find the derivative of (i) (x^5 − cosx)/sinx Let f(x) = (x^5 − cosx)/sinx Let u = x5 – cos x & v = sin x So, f(x) = (𝑢/𝑣) ∴ f’(x) = (𝑢/𝑣)^′ Using quotient rule f’(x) = (𝑢^′ 𝑣 −〖 𝑣〗^′ 𝑢)/𝑣^2 Finding u’ & v’ u = x5 – cos x u’ = 5. x5 – 1 – ( – sin x) = 5x4 + sin x v = sin x v’ = cos x Now, f’(x) = (𝑢^′ 𝑣 −〖 𝑣〗^′ 𝑢)/𝑣^2 Derivative of xn is nxn – 1 & Derivative of cos x = – sin x (Derivative of sin x = cos x) = ((5x4 + sin〖x) sin x −(cos x)(x5 − cos x) 〗)/sin2x = (5x4 sin〖x + sin 2x − cos x . x5 + cos2 x〗)/(sin2 x) = (−x5 cos〖x + 5x4 sinx + 𝐬𝐢𝐧𝟐 𝐱 + 𝐜𝐨𝐬𝟐 𝐱〗)/(sinx )2 = (−x5 cos〖x + 5x4 sinx + 𝟏〗)/(sinx )2 Thus, f’(x) = (−𝐱𝟓 𝐜𝐨𝐬〖𝐱 + 𝟓𝐱𝟒 𝐬𝐢𝐧𝒙 + 𝟏〗)/(𝐬𝐢𝐧𝐱 )𝟐 (Using sin2x + cos2x = 1)