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Suppose that functions f and g and their derivatives have the following values at x = 2 and x = 3. x 2 f x 8 g x 2 f x 1/3 g x –3 5 2 Evaluate the derivatives with respect to x of the following combinations at the given value of x. 3 (a) 2 f d dx x 2 f 3 –4 at x = 2 At x = 2: x 2 f x 1 2 2 f 2 2 3 3 Suppose that functions f and g and their derivatives have the following values at x = 2 and x = 3. x 2 f x g x 8 2 f x 1/3 g x –3 5 2 Evaluate the derivatives with respect to x of the following combinations at the given value of x. 3 (b) f d dx –4 3 x g x at x = 3 f x g x f x g x At x = 3: f 3 g 3 2 5 Suppose that functions f and g and their derivatives have the following values at x = 2 and x = 3. x 2 f x g x 8 2 f x 1/3 g x –3 5 2 Evaluate the derivatives with respect to x of the following combinations at the given value of x. 3 (c) f d dx x f 3 –4 g x at x = 3 x g x f x g x g x At x = 3: f x f 3 g 3 g 3 f 3 3 5 4 2 1 5 8 Suppose that functions f and g and their derivatives have the following values at x = 2 and x = 3. x 2 f x g x 8 f x 2 g x –3 1/3 5 2 Evaluate the derivatives with respect to x of the following combinations at the given value of x. 3 –4 3 g x at x = 2 g 2 f 2 f 2 g2 d f 2 g 2 2 dx g 2 (d) f x 2 1 3 8 3 2 2 74 3 4 37 6 Suppose that functions f and g and their derivatives have the following values at x = 2 and x = 3. x 2 f x 8 g x 2 f x 1/3 g x –3 5 2 Evaluate the derivatives with respect to x of the following combinations at the given value of x. 3 (e) f d dx g x 3 –4 at x = 2 f g 2 f g 2 g 2 f 2 g 2 1 3 1 3 Suppose that functions f and g and their derivatives have the following values at x = 2 and x = 3. x f 2 x g x 8 2 f x g x –3 1/3 5 2 Evaluate the derivatives with respect to x of the following combinations at the given value of x. 3 (f) f d dx x –4 3 at x = 2 1 f 2 2 d f 2 dx f 2 f 2 2 f 2 1 3 2 8 1 12 2 Suppose that functions f and g and their derivatives have the following values at x = 2 and x = 3. f x 2 x 8 g x f x 2 g x –3 1/3 5 2 Evaluate the derivatives with respect to x of the following combinations at the given value of x. 3 (g) 1 g d dx 2 x 3 –4 at x = 3 2 g 3 2 g 3 3 d dx g 3 2 g 3 g 3 2 5 4 3 5 32 3 Suppose that functions f and g and their derivatives have the following values at x = 2 and x = 3. f x 2 x g x 8 2 f x 1/3 g x –3 5 2 Evaluate the derivatives with respect to x of the following combinations at the given value of x. 3 (h) d dx f 2 f x g x 2 2 2 at x = 2 2 g 2 2 d d 2 f 2 f 2 2g 2 g 2 2 2 dx dx f 2 g 2 1 –4 3 Suppose that functions f and g and their derivatives have the following values at x = 2 and x = 3. f x 2 x g x 8 f x 2 1/3 g x –3 5 2 Evaluate the derivatives with respect to x of the following combinations at the given value of x. 3 (h) x g x at x = 2 2 f 2 g 2 g2 f f –4 3 2 2 f 2 2 g 2 8 1 3 2 3 8 2 2 2 10 3 68 10 3 2 17 2 5 3 17 Slopes of Parametrized Curves A parametrized curve (x(t), y(t)) is differentiable at t if x and y are differentiable at t. At a point on a differentiable parametrized curve where y is also a differentiable function of x, the derivatives dy/dt, dx/dt, and dy/dx are related by the Chain Rule: dy dt dy dx dx dt Usually, we write this in a different form… If all three derivatives exist and d x d t 0 , dy dx dy dt dx dt Practice Problems Find the equation of the line tangent to the curve at the point defined by the given value of t. x sin 2 t y cos 2 t t 1 6 Find the three derivatives: dx dt dy dt dy dx cos 2 t d dt sin 2 t dy dt dx dt 2 t d dt 2 co s 2 t 2 t 2 sin 2 t 2 sin 2 t 2 cos 2 t tan 2 t Practice Problems Find the equation of the line tangent to the curve at the point defined by the given value of t. x sin 2 t y cos 2 t t 1 6 2 2 3 1 The line passes through: sin , cos , 6 6 2 2 2 And has slope: tan 3 6 Equation of the tangent line: y 1 2 3 3x 2 y 3x 2 Practice Problems Find the equation of the line tangent to the curve at the point defined by the given value of t. x 2t 3 yt 2 Derivatives: dx dt 4t dy 4t dt Point: 2 1 3, 1 2 4 3 t 1 dy dx 4 5,1 dy dt dx dt t 4t Slope: 1 1 2 Equation of the tangent line: y 1 1 x 5 4t 3 y x4 2