Question

3. Find the equation of the tangent line to the curve 2x^3 + y^2 = xy at the point (−1, 1).

4. Use implicit differentiation to find y' for sin(xy^2 ) − x^3 = 4x + 2y.

5. Use logarithmic differentiation to find y' for y = e^4x cos(2x) / (x−1)^4 .

6. Show that d/dx (tan (x)) = sec^2 (x) using only your knowledge of the derivatives of sine/cosine with derivative rules.

7. Use implicit differentiation to show that d/dx (tan−1 (x)) = 1 / 1+x^2 .

Answer #1

1. Find the equation of the tangent line to the graph of ?2? −
5??2 + 6 = 0 at (3,1).
2.. Find the equation of the normal line to the graph of sin(??) =
? at the point (?/2 ,1).
3.. Find the equation of the tangent line to the graph of cos(??) =
? at the point (0.1)
Find implicit differentiation of dy/dx
a) xy=x+y
b) xcosy=y
c)x^3 +y^2=0

Use implicit differentiation to find an equation of the line
tangent to the curve sin(x+y)=2x-y at the point (pi,2\pi )

Use implicit differentiation to find an equation of the line
tangent to the curve sin(x+y)=2x-y at the point (pi, 2pi )

A) Use implicit differentiation to find an equation of the
tangent line to the ellipse defined by
5x^2+4xy+3y^2=12 at the point (−1,−1)
B) Find dy/dx by implicit differentiation, if ey=2x−2y

3. Consider the equation:
x^2y −√y = 2 + 4x^2
a) Find dy/dx using implicit differentiation. b)Construct the
equation of the tangent line to the graph of this equation at the
point (1, 9)

Differentiate the function
y=ln(e-x +xe-x)
Find y and y"
y=ln(sec(3x)+tan(3x))
Use logarithmic differentiation to find the derivative
of the function.
y=(cos(9x))x
Use logarithmic differentiation to find the derivative
of the function.
y=(sin(9x))(lnx)

4) Use implicit differentiation to find the equation of the
tangent line to the curve xy^3+xy=16 at the point (8,1). The
equation of this tangent line can be written in the form
y=mx+by=mx+b where m is:
and where b is:

If 5x^2+3x+xy=3 and y(3)=-17, find y'(3) by implicit
differentiation.
Thus an equation of the tangent line to the graph at the point
(3,-17) is

(a) Find an equation of the plane tangent to the surface xy ln x
− y^2 + z^2 + 5 = 0 at the point (1, −3, 2)
(b) Find the directional derivative of f(x, y, z) = xy ln x −
y^2 + z^2 + 5 at the point (1, −3, 2) in the direction of the
vector < 1, 0, −1 >. (Hint: Use the results of partial
derivatives from part(a))

Write an equation for the tangent line to the plane curve xy^2 −
2x^2 + 3 x + y = 7 at the point P .(1,-3)

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