Find the derivative of y with respect to x if 2^xy=1 . Show all your work.
To find the derivative of y with respect to x, we will use logarithmic differentiation.
Taking the natural logarithm of both sides of the equation, we get:
ln(2^xy) = ln(1)
Using the logarithm properties, we can rewrite this as:
xy * ln(2) = ln(1)
ln(1) is equal to 0, so we can simplify further:
xy * ln(2) = 0
Now, we will differentiate both sides of the equation with respect to x.
Using the product rule for differentiation, the left side becomes:
d/dx [xy * ln(2)] = d(xy)/dx * ln(2) + xy * d(ln(2))/dx
Since y is a function of x, we can differentiate xy with respect to x as:
d(xy)/dx = y + x * dy/dx
The derivative of ln(2) with respect to x is 0, since ln(2) is a constant.
So, the equation becomes:
y + x * dy/dx * ln(2) = 0
To isolate dy/dx, we can move the terms involving dy/dx to the other side of the equation:
x * dy/dx * ln(2) = -y
Now, we can solve for dy/dx by dividing both sides of the equation by x * ln(2):
dy/dx = -y / (x * ln(2))
Therefore, the derivative of y with respect to x is:
dy/dx = -y / (x * ln(2))