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. Derivative :Chain Rule (Examples List) 1. Find dy if dx (a) y = u12 and u = 5x2 3x + 2 p x+2 (b) y = u and u = x 2 p u+2 (c) y = and u = x u 2 x2 u (d) y = e and u = 2 2. Find the following derivatives. dy x if y = sin u and u = dx 3 dv (b) if v = cos s and s = et dt dE 1 (c) if w = and z = q 2 + R2 ; where R is a positive constant. dq z (a) 3. Use the chain rule to …nd the derivative of y = (sin x + cos x)5 . Do it in two ways: (a) by introducing a “middle”variable u; and (b) by using the “outer-inner”procedure. 4. Find the derivative of the following functions. f (x) = p 3 x4 + x2 + 1; g (x) = x 2 x +1 12 q p ; h (x) = x + x: 5. Di¤erentiate the following functions. R ( ) = cos (4 ) ; F (t) = sin (sin t) ; G (w) = tan (5ew ) ; y = sec (2x 3) : 6. Find the derivative of the following functions. y = sin2 x; z = tan5 x; w = sec3 x; u = ekx (where k is a non-zero constant) 7. Di¤erentiate the following functions and simplify your answers. y = x2 + 3x 2 e x ; y = x2 sin 3 1 x 8. Di¤erentiate y = (2x + 5)4 (x2 + 1) and factor your answer. x 9. Di¤erentiate f (x) = p and simplify your answer. 4 x2 10. For f (x) = (x2 + 5x + 4) e3x ; …nd f 0 (0) and f 00 (0) : 11. Find the third derivative of y = tan2 x: 1 12. Find an equation of the tangent line to the graph of y = p x3 + 1 at the point (2; 3) : t 13. Suppose that the position of a moving particle is given by s (t) = e t sin ; where s is 2 in meters and t in seconds. Find its velocity and acceleration at time t = 1: 14. Suppose that air is being pumped into a spherical balloon so that its volume V is increasing at a constant rate of 36 cm3 per second. How fast is its radius r increasing 4 3 at the moment when its radius reaches 3 cm? [Volume of a sphere: V = r :] 3 p 15. If g is a di¤erentiable function and f (x) = x2 + 4g (x); how is f 0 (x) related to g (x) and g 0 (x)? Find f 0 (3) if g (3) = 4 and g 0 (3) = 5: 16. If g is a di¤erentiable function and f (t) = g (sin 3t) ; how is f 0 related to g 0 ? Find f 0 (0) if g 0 (0) = 4: 17. Suppose that f and g are di¤erentiable functions and some of the values of f; g; f 0 and g 0 are known to us: x 1 2 3 4 f (x) 2 4 1 3 g (x) 3 1 4 2 0 f (x) 5 11 3 7 g 0 (x) 4 9 6 10 Find (f g)0 (2), (g f )0 (4) ; d (f (2x)) dx and x=1 dp g (x) dx dy = ky; show that y = e 18. If y = f (x) satis…es the equation dx derivative. Joseph Ling (Fall 2013) 2 : x=3 kx f (x) has constant zero