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G radients of Inverse Trig Functions Use the relationship

G radients of Inverse Trig Functions Use the relationship . Ex y = sin –1 x This is the same as siny = x ie sin both sides So x = siny. Differentiate this expression. Remember the letter that comes 1 st goes on top The letter that comes 2 nd goes underneath. x = siny.

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G radients of Inverse Trig Functions Use the relationship

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  1. Gradients of Inverse Trig Functions Use the relationship Ex y = sin–1x This is the same as siny = x iesin both sides So x = siny Differentiate this expression. Remember the letter that comes 1st goes on top The letter that comes 2nd goes underneath.

  2. x = siny Using sin2x + cos2x = 1 replace y by x. So sin2y + cos2y = 1 Make cosy the subject cos2y = 1 – sin2y But x = siny So

  3. This technique can be used to differentiate y = cos–1x and y = tan–1x (use 1 + tan2x = sec2x) Find the gradient of y = cos–1x and y = tan–1x

  4. Differentiating Hyperbolic Functions y = sinh x = • y = cosh x = • y = tanh x = Prove it using the quotient rule

  5. Using techniques from C4 (DIFIU) y = sinh 2x y = cosh x y = tanh (3x+2)

  6. Differentiating Inverse Hyperbolic Functions • y = sinh–1x so x = sinhy cosh2x – sinh2x = 1 Using Osbornes Rule

  7. Differentiating Inverse Hyperbolic Functions • y = cosh–1x so x = coshy cosh2x – sinh2x = 1 Using Osbornes Rule

  8. Differentiating Inverse Hyperbolic Functions • y = tanh–1x so x = tanhy 1 – tanh2x = sech2x Using Osbornes Rule See FP2 Notes for further examples

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