__Category__: Differential Calculus, Algebra

"Published in Vacaville, California, USA"
Find a formula for

__Solution__:
The first thing that we have to do is to get the derivative of uv with respect to x where u and v are functions of x. Apply the derivative by product formula, we have

Take the derivative again of the above equation with respect to x by product formula, we have

where:
__Category__: Differential Calculus, Analytic Geometry, Algebra, Trigonometry

"Published in Newark, California, USA"

Find the angle of intersection between the two curves:
__Solution__:
Consider the given pair of two curves above
The first thing that we need to do is to get their point of intersection. If you subtract the first equation from the second equation, the left side of the equation will be equal to zero. Hence, the resulting equation is
The value of y is
The point of intersection of the given two curves is P(1, ½).
Consider the first given equation of a curve
Take the derivative on both sides of the equation with respect to x by implicit differentiation, we have
The
slope of a curve is equal to the first derivative of the equation of a
curve with respect to x. In this case, dy/dx is the slope of a curve.
To
get the value of the slope of a curve at their point of intersection,
substitute x = 1 and y = ½ at the equation above, we have
The
slope of a curve at their point of intersection is equal to the slope
of tangent line that passes thru also at their point of intersection.
Hence,
Consider the second given equation of a curve
Take the derivative on both sides of the equation with respect to x by implicit differentiation, we have

The
slope of a curve is equal to the first derivative of the equation of a
curve with respect to x. In this case, dy/dx is the slope of a curve.
To
get the value of the slope of a curve at their point of intersection,
substitute x = 1 and y = ½ at the equation above, we have
The
slope of a curve at their point of intersection is equal to the slope
of tangent line that passes thru also at their point of intersection.
Hence,
Therefore, the angle between two curves at their point of intersection is

or