mathematics, precalculus

Question

Find 𝑓_x and 𝑓_y for the given functions: a) 𝑓(𝑥, 𝑦) = ln√x²+y² b) 𝑓(𝑥, 𝑦) = e^xylny c) 𝑓(𝑥, 𝑦) = x ⁄ x²–xy²

Answer 1

I'll go through some of these to give you a better understanding

Let's get f_x for a).

We'll use the chain rule. The derivative of ln(x) is 1/x.

f_x = 1/√(x² + y²) • d/dx(√(x² + y²))

Use the chain rule again. The derivative of √x is 1/(2√x)

f_x = 1/√(x² + y²) • 1/(2√x² + y²) • d/dx(x² + y²)

Use the chain rule again. The derivative of x² + y² with respect to x is 2x because y² is seen as a constant.

Therefore,

f_x = 1/√(x² + y²) • 1/(2√x² + y²) • 2x.

Let's get f_y for b).

We have two functions multiplied both with y in them, so we'll have to use the product rule. If we were taking the derivative with respect to x, we wouldn't need the product rule because ln y would be seen as a constant.

Applying the product rule we get:

e^xy • d/dy(ln y) + (ln y) • d/dy (e^xy)

e^xy(1/y) + (ln y)(xe^xy)

Notice that the derivative of e^xy with respect to y is xe^xy because x is seen as a constant.