Post....whatever!

So, Let's go with implicit differentiation...just solve for dy / dx.


Example 1: Use implicit differentiation to find the derivative dy / dx where y x + sin y = 1

Solution to Example 1:



Use the sum rule of differentiation to the whole term on the left of the given equation.

d [xy] / dx + d [siny] / dx = d[1]/dx .

Differentiate each term above using product rule to d [xy] / dx and the cain rule to d [siny] / dx.

x dy / dx + y + (dy / dx) cos(y) = 0 .


Note that in calculating d [siny] / dx, we used the chain rule since y is itself a function of x and sin (y) is a function of a function.


Solve for dy/dx to obtain.

dy / dx = -y / (x + cos y)

Example 2: Use implicit differentiation to find the derivative dy / dx where y 4 + x y 2 + x = 3

Solution to Example 2:


Use the differentiation of a sum formula to left side of the given equation.

d[y 4] / dx + d[x y 2] / dx + d[x] / dx = d[3] / dx


Differentiate each term above using power rule, product rule and chain rule.

4y 3 dy / dx + (1) y 2 + x 2y dy / dx + 1 = 0


Solve for dy/dx.

dy/dx = (-1 - y 2) / (4y 3 + 2xy)


Example 3: Find all points on the graph of the equation


x 2 + y 2 = 4


where the tangent lines are parallel to the line x + y = 2

Solution to Example 3:


Rewrite the given line x + y = 2 in slope intercept form: y = -x + 2 and identify the slope as m = -1. The tangent lines are parallel to this line and therefore their slope are equal to -1. The slope of tangent lines at a point can be found by implicity differentiation of x 2 + y 2 = 4

2x + 2y dy/dx = 0


Let P(a , b) be the point of tangency. At point P the slope is -1. Substituting x by a, y by b and dy/dx by -1 in the above equation, we obtain

2a + 2b (-1) = 0


Point P(a , b) is on the graph of x 2 + y 2 = 4, hence

a 2 + b 2 = 4


Solve the system of equations: 2a - 2b = 0 and a 2 + b 2 = 4 to obtain two points

(-sqrt(2) , -sqrt(2)) and (sqrt(2) , sqrt(2))



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