[nickie6038]

Use implicit differentiation to find y'
ln(xy) = x+y


Use implicit differentiation to find y' at (5,2)
3xy + 3x = 45


Find the differential dy given the following function if p and q are constants
f(x) = x^p + x^q


Does the following function have an inverse on the interval (2,5)?
f(x) = (x-1)/(x^2-1)


Let g(x) be the inverse function of f(x). Find the equation of the line tangent to the curve g(x)at the point (-19,5)
f(x) = x^2 - 8x -4

[nightal1031]

do you just have a problem finding derivatives?

to find the derivative of a function implicitly, you take the derivative of both sides of the equation with respect to x. you then solve the equation for dy/dx.

ln(x+y) = x+y
ln x + ln y = x+y (all I did here was use a logarithmic property to separate x and y)
1/x + 1/y(dy/dx) = 2 (I took the derivative of both sides with respect to x)
1/y (dy/dx) = 2- (1/x) (now I'm solving for dy/dx)
dy/dx = y* (2x/x -1/x) (getting a common denominator)
dy/dx = (2xy-y)/x


for the second question, find the derivative of the function implicitly as I showed you above, but then sub 5 in for x and 2 in for y after you have found the derivative. I'm not going to show the whole process, but I got -3 as an answer.

for the third question, take the derivative of the function
dy/dx = px^(p-1) + qx^ (q-1)

then solve for dy by multiplying by dx.
dy= dx[px^(p-1) + qx^(q-1)]


for the last two, I can't remember how to find the inverse function. The only thing I can remember about inverse functions is that if you graph both equations, you will have a line of symmetry between the two on the equation y=x. I hope I at least helped with the first 3 questions!

[vrooje]

Inverse functions are symmetric about the line y=x, true.

That means that you can find the inverse function by swapping the x and y variables.

So if you have , you can write this as and then find the inverse by swapping x and y:



Then differentiate that with respect to x and find the slope of the tangent line and then the equation of the tangent line using the point they gave you.

For the other question about inverses, they're probably really asking if the inverse is a function -- and a more quick way to check this than finding the inverse may be to use the horizontal line test. The vertical line test checks to see if a given curve is a function, and the horizontal line test checks to see if the inverse of a curve is a function.