Find out the partial derivative w.r.t 'x' and 'y' of
f (x,y) = log(y) x
Now,
log(y) x = ln x / ln y
Partial Diff w.r.t 'x' = 1/ x ln y
so can you find out what will be the partial derivatives w.r.t 'y'
Bbxjxj
To find the partial derivative with respect to 'y' of f(x, y) = log(y) * x, we will differentiate the expression with respect to 'y' while treating 'x' as a constant.
Using the chain rule, the partial derivative with respect to 'y' can be found as follows:
∂/∂y [log(y) * x] = (∂/∂y(log(y))) * x
To differentiate log(y) with respect to 'y', we can use the logarithmic differentiation rule:
d/dy [log(y)] = 1/y
Therefore, the partial derivative with respect to 'y' is given by:
∂/∂y [log(y) * x] = (1/y) * x
Hence, the partial derivative with respect to 'y' of f(x, y) = log(y) * x is (1/y) * x.
To find the partial derivative with respect to 'y', we will differentiate the function with respect to 'y' while treating 'x' as a constant.
Given: f(x, y) = (ln(x) / ln(y)).
To find ∂f/∂y, we will differentiate the function f with respect to 'y' while treating 'x' as a constant.
Start by applying the quotient rule for differentiation:
∂f/∂y = (d/dy)[ln(x)] / ln(y) - [ln(x)](d/dy)[ln(y)] / [ln(y)]^2
Since 'x' is treated as a constant, the derivative of ln(x) with respect to 'y' is 0.
∂f/∂y = 0 - [ln(x)](1/y) / [ln(y)]^2
So, the partial derivative of f with respect to 'y' is:
∂f/∂y = -ln(x) / (y ln(y))^2
Therefore, the partial derivative with respect to 'y' of f(x, y) = (ln(x) / ln(y)) is equal to -ln(x) / (y ln(y))^2.