Find the partial derivatives of f(x,y)=(sinx + y)^(y sinx)
To find the partial derivatives of f(x, y) = (sinx + y)^(y sinx), we need to differentiate the function with respect to each variable separately. Let's start with the partial derivative with respect to x, denoted as ∂f/∂x.
To find ∂f/∂x, we differentiate f(x, y) with respect to x while treating y as a constant. Note that the function (sinx + y)^(y sinx) involves a composition of functions, namely sine and exponentiation. We can apply the chain rule to differentiate it properly.
Let's break down the steps:
Step 1: We differentiate the outer function while holding the inner function constant. In this case, the outer function is u^v, where u = sinx + y and v = y sinx. The derivative of u^v with respect to u is v * (u^(v-1)). So, we differentiate (sinx + y) with respect to u, treating v as a constant.
du/dx = cosx (derivative of sinx) + 0 (derivative of y with respect to x, treated as a constant) = cosx
Step 2: We differentiate the inner function v = y sinx with respect to x.
dv/dx = y * (derivative of sinx) + sinx * (derivative of y with respect to x) = y * cosx + sinx * 0 = y * cosx
Step 3: Finally, we apply the chain rule by multiplying the results from Step 1 and Step 2.
∂f/∂x = v * (u^(v-1)) * (du/dx) = (y sinx) * ((sinx + y)^(y sinx - 1)) * cosx = y cosx * (sinx + y)^(y sinx - 1) * sinx
So, we have found the partial derivative ∂f/∂x.
To find the partial derivative with respect to y, denoted as ∂f/∂y, the process is similar, but this time we treat x as a constant.
Step 1: We differentiate the outer function while holding the inner function constant. In this case, the outer function is u^v, where u = sinx + y and v = y sinx. The derivative of u^v with respect to u is v * (u^(v-1)). So, we differentiate (sinx + y) with respect to u, treating v as a constant.
du/dy = 1 (derivative of y with respect to y) + 0 (derivative of sinx with respect to y, treated as a constant) = 1
Step 2: We differentiate the inner function v = y sinx with respect to y.
dv/dy = y * 0 + sinx * 1 (derivative of sinx with respect to y, treated as a constant) = sinx
Step 3: Finally, we apply the chain rule by multiplying the results from Step 1 and Step 2.
∂f/∂y = v * (u^(v-1)) * (du/dy) = (y sinx) * ((sinx + y)^(y sinx - 1)) * 1 = y sinx * (sinx + y)^(y sinx - 1)
So, we have found the partial derivative ∂f/∂y.
To summarize:
∂f/∂x = y cosx * (sinx + y)^(y sinx - 1) * sinx
∂f/∂y = y sinx * (sinx + y)^(y sinx - 1)