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Implicit Differentiation to Find Derivative

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Consider the curve xy^2 + y = 3 - x^2. Use implicit differentiation to find the derivative dy/dx

Hence determine the tangent to the curve at the point (-2,1)

A stone dropped into a still pond sends out a circular ripple whose radius increases at a constant rate of 0.5ms-1. How rapidly is the area enclosed by the ripple increasing when the radius has become 5m?

Hint: This is a related rates question. A suggested start is to assign letters to all quantities that vary and then identify the known rates and the rate that is to be found.

Graph on the same axis:

1. y = 1/x

2. y = 1/x-3

3. y = 2 + (1/x-3)

Clearly label key points (eg. turning points, end points, discontinuities, etc if these exist).

Additionally, determine the domain and range for all four functions.

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Finding a derivative by implicit differentiation

Question 3:
a) Find the derivative of the following functions dy/dx, by the method of implicit differentiation: i) (x^2)(y^2) - xy = 8; ii) sqrt(x + y) = x; iii) (x - y)/(2x + 3y) = 2x.

b) Find the equation of tangent line to the graph of the following function at the indicated points: i) y^2 - x^2 = 16; at (2, 2 sqrt(5); ii) (x^2)(y^3) - y^2 + xy - 1 = 0 at (1, 1).

Question 4:
a) Find the derivative of the following functions: i) f(x) = 4e^(3x+2) ... ii) f(x) = (x^3)(e^x); iii) f(x) = (x-1)e^3x+2; iv_ Find the second derivative of the function: f(x) = 2xe^3x.

b) i) find an equation of the tangent line to the graph of the function f(x) = xln, at; (1,1). ii) Find the equation of the tangent line to the graph of y = e^2x-3 at (3/2, 1).

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