Answer the following. Write your answer on a separate sheet of paper. 1. Differentiate y = x2 sin x 2. What is the derivative of y = 3 sin x - 4 cos x? 3. By applying the product rule, differentiate y = x3 tanx. 4. Solve the derivative of y = csc x cot x by applying the product rule. 5. y = (sinx) 2 (cosx)?) what is y'? Solve for the derivative of the following functions. Write your answer on a separate sheet of paper. 6. y = x sin(2x) 11. y = csc(2x2) 7. y = cos(x2) 12. y = csc(3x2 + 1) 8. y = cos2x 13. y = 2 sec x + 50 9. y = tan(sinx) 14. y = -3 sec(6x) 10.y = tan(4x - 1) 15. y = 3 cot(5x)What is It Rules of Derivatives of Trigonometric Functions Rule 1. - (sinu) = cosu W Example. Differentiate y = sin 4x. Solution: y = sin 4x Given ay = cos(4x) (4x) Chain rule has been applied here. ay = cos(4x) . 4. (x) Differentiation is linear. Differentiated them separately and pulled out constant factor = 4 cos 4x . 1 Differentiated the constant 1. dx Final answer. dx = 4 cos 4x Rule 2. d (cosu) =-sinuz Example. Differentiate y = cos(2x). Solution: y = cos(2x) Given a= - sin(2 x) -(2x) Chain rule has been applied here. dy dx = -2 . " (x) . sin(2x) Differentiation is linear. Differentiated them separately and pulled out constant factor = -2 . 1 sin(2x) Derivative of x is 1. -= -2 sin(2x) Final answer. Rule 3. ~ (tanu) = seczu du Example. Differentiate y = tan(2x). Solution: y = tan(2x) Given dx dy = sec2 (2x) - (2x) Chain rule has been applied here . ay = sec2 (2x) . 2 . (x) Differentiation is linear. Differentiated them separately and pulled out constant factor "y = 2 sec2 (2x) . 1 Derivative of x is 1.dy = 2 secz (2x) Final answer. Rule 4. " (cotu) = -cschu 7 Example. Differentiate y = 4x2 + cotx. Solution: y = 4x4 + cotx Given (cotx) +4 . "dy ( x 2 ) Differentiation is linear. Differentiated them separately and pulled out constant factor " = (-csc2x) + 4 . 2x Applied the differentiation rule for cot x and applied the power rule for x2 dx dy = 8x - csc x Final answer. Rule 5. ~ (secu) = secu tanu-" Example. Differentiate y = sec(2x). Solution: y = sec(2x) Given 2 = sec(2x) tan (2x) (2x) Chain rule has been applied here. dx = sec(2x) tan(2x) . (2) . (x) Differentiation is linear. Differentiated them separately and pulled out constant factor dy Derivative of x is 1. dx = 2 sec (2x) tan (2x) . 1 ay = 2 sec (2x) tan (2x) Final answer Rule 6. ~ (cscu) = - cscucotu" Example. Differentiate y = csc(5x). Solution: y = csc(5x) Given dy = - csc(5x) cot(5x) ~ (5x) Chain rule has been applied here. ay = - csc(5x) cot(5x) . 5 . ~ (x) Differentiation is linear. Differentiated them separately and pulled out constant factor ay = -5 cot(5x) csc(5x) . 1 Derivative of x is 1. dx dy -5 cot(5x) csc(5x) Final