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(c) y = frac{e^{2x}-1}{e^{2x}+1} (d) y = (e^{3x}+1) ln x (e) y = e^{sqrt{x}}(1+sqrt{x}) (f) y = (e^{3x}+ln(3x))sin(3x) 6. Find y'' and y''' for the following functions. (a) y = 7x^{4} - 5x^{3} - 2x (b) y = (x^{3}+5)(3x+2) 7. Verify that the following functions satisfy the given equation where A and B are constants. (a) y = Ax + Bx^{2}, quad x^{2}frac{d^{2}y}{dx^{2}} - 2xfrac{dy}{dx} + 2y = 0. (b) y = Ax^{3} + frac{B}{x^{3}}, quad x^{2}frac{d^{2}y}{dx^{2}} + xfrac{dy}{dx} - 9y = 0. 8. Use implicit differentiation to find frac{dy}{dx} of the following (a) x^{3} - xy + y^{2} = 7 (b) e^{2x} = sin(x + 3y) (c) xy = cot(xy) (d) y^{2}cosleft(frac{1}{y} ight) = 2x + 2y (e) y = a^{x^{3}} (f) y = x^{2}(x^{2}-1)^{3}(x+1)^{4} 9. Use implicit differentiation to find frac{dy}{dx} and then left.frac{d^{2}y}{dx^{2}} ight|_{(x_{0},y_{0})} of the following equations (a) x^{2/3} + y^{2/3} = 1, quad P(1,-1) (b) 2y^{2} - 2xy = 2 + x, quad P(0,1) 10. Find the frac{dy}{dx} and frac{d^{2}y}{dx^{2}} for the curves with the following parametric equations. (a) x = 3t^{2} - 3t^{3}, quad y = t(t+3) (b) x = frac{t}{t-1}, quad y = t^{2} - 1 (c) x = 3sin t, quad y = e^{t} (d) x = 5cos^{3}t, quad y = 7sin^{3}t 11. Assuming that the following equations define x and y implicitly as differentiable functions. Find tangent line equation to the curve x = f(t), y = g(t), at a given value of t. (a) x(t) = 4 - sqrt{t}, quad y(t) = sqrt{t} + 1, quad t = 9. (b) xsin t + 2x = t, quad tsin t - 2t = y, quad t = pi. (c) x^{2} - 2tx + 2t^{2} = 4, quad 2y^{3} - 3t^{2} = 4, quad t = 2. (d) x + 2x^{3/2} = t^{2} + t, quad ysqrt{t+1} + 2tsqrt{y} = 4, quad t = 0.

          (c) y = frac{e^{2x}-1}{e^{2x}+1}
(d) y = (e^{3x}+1) ln x
(e) y = e^{sqrt{x}}(1+sqrt{x})
(f) y = (e^{3x}+ln(3x))sin(3x)

6. Find y'' and y''' for the following functions.
(a) y = 7x^{4} - 5x^{3} - 2x
(b) y = (x^{3}+5)(3x+2)

7. Verify that the following functions satisfy the given equation where A and B are constants.
(a) y = Ax + Bx^{2}, quad x^{2}frac{d^{2}y}{dx^{2}} - 2xfrac{dy}{dx} + 2y = 0.
(b) y = Ax^{3} + frac{B}{x^{3}}, quad x^{2}frac{d^{2}y}{dx^{2}} + xfrac{dy}{dx} - 9y = 0.

8. Use implicit differentiation to find frac{dy}{dx} of the following
(a) x^{3} - xy + y^{2} = 7
(b) e^{2x} = sin(x + 3y)
(c) xy = cot(xy)
(d) y^{2}cosleft(frac{1}{y}
ight) = 2x + 2y
(e) y = a^{x^{3}}
(f) y = x^{2}(x^{2}-1)^{3}(x+1)^{4}

9. Use implicit differentiation to find frac{dy}{dx} and then left.frac{d^{2}y}{dx^{2}}
ight|_{(x_{0},y_{0})} of the following equations
(a) x^{2/3} + y^{2/3} = 1, quad P(1,-1)
(b) 2y^{2} - 2xy = 2 + x, quad P(0,1)

10. Find the frac{dy}{dx} and frac{d^{2}y}{dx^{2}} for the curves with the following parametric equations.
(a) x = 3t^{2} - 3t^{3}, quad y = t(t+3)
(b) x = frac{t}{t-1}, quad y = t^{2} - 1
(c) x = 3sin t, quad y = e^{t}
(d) x = 5cos^{3}t, quad y = 7sin^{3}t

11. Assuming that the following equations define x and y implicitly as differentiable functions. Find tangent line equation to the curve x = f(t), y = g(t), at a given value of t.
(a) x(t) = 4 - sqrt{t}, quad y(t) = sqrt{t} + 1, quad t = 9.
(b) xsin t + 2x = t, quad tsin t - 2t = y, quad t = pi.
(c) x^{2} - 2tx + 2t^{2} = 4, quad 2y^{3} - 3t^{2} = 4, quad t = 2.
(d) x + 2x^{3/2} = t^{2} + t, quad ysqrt{t+1} + 2tsqrt{y} = 4, quad t = 0.

$(c) y = frace^2x-1e^2x+1 (d) y = (e^3x+1) ln x (e) y = e^sqrtx(1+sqrtx) (f) y = (e^3x+ln(3x))sin(3x) 6. Find y” and y”' for the following functions. (a) y = 7x^4 - 5x^3 - 2x (b) y = (x^3+5)(3x+2) 7. Verify that the following functions satisfy the given equation where A and B are constants. (a) y = Ax + Bx^2, quad x^2fracd^2ydx^2 - 2xfracdydx + 2y = 0. (b) y = Ax^3 + fracBx^3, quad x^2fracd^2ydx^2 + xfracdydx - 9y = 0. 8. Use implicit differentiation to find fracdydx of the following (a) x^3 - xy + y^2 = 7 (b) e^2x = sin(x + 3y) (c) xy = cot(xy) (d) y^2cosleft(frac1y ight) = 2x + 2y (e) y = a^x^3 (f) y = x^2(x^2-1)^3(x+1)^4 9. Use implicit differentiation to find fracdydx and then left.fracd^2ydx^2 ight|(x0,y0) of the following equations (a) x^2/3 + y^2/3 = 1, quad P(1,-1) (b) 2y^2 - 2xy = 2 + x, quad P(0,1) 10. Find the fracdydx and fracd^2ydx^2 for the curves with the following parametric equations. (a) x = 3t^2 - 3t^3, quad y = t(t+3) (b) x = fractt-1, quad y = t^2 - 1 (c) x = 3sin t, quad y = e^t (d) x = 5cos^3t, quad y = 7sin^3t 11. Assuming that the following equations define x and y implicitly as differentiable functions. Find tangent line equation to the curve x = f(t), y = g(t), at a given value of t. (a) x(t) = 4 - sqrtt, quad y(t) = sqrtt + 1, quad t = 9. (b) xsin t + 2x = t, quad tsin t - 2t = y, quad t = pi. (c) x^2 - 2tx + 2t^2 = 4, quad 2y^3 - 3t^2 = 4, quad t = 2. (d) x + 2x^3/2 = t^2 + t, quad ysqrtt+1 + 2tsqrty = 4, quad t = 0.$

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