partial derivative from **** problem

[goaway716]

If f_x and f_y are the partial derivatives of f(x,y) and f_s and f_t are the partial derivatives of f viewed as a function of s and t, show that if:

s=(x^3)-(3xy²)

t=3x²y-y^3

then

xf_x+yf_y=3(sf_s+tf_t)

please help me finishing this problem as i got no clue how to go about it
thanx

 

[Deleted member 4993]

If f_x and f_y are the partial derivatives of f(x,y) and f_s and f_t are the partial derivatives of f viewed as a function of s and t, show that if:

s=(x^3)-(3xy²)

t=3x²y-y^3

then

xf_x+yf_y=3(sf_s+tf_t)

please help me finishing this problem as i got no clue how to go about it
thanx

Start with

\(\displaystyle \frac{df}{dx} \ = \ \frac{df}{ds}\cdot \frac{ds}{dx} \ + \ \frac{df}{dt}\cdot \frac{dt}{dx}\)

All the derivatives above are partial derivative

Please share your work with us, indicating exactly where you are stuck - so that we may know where to begin to help you.

 

[galactus]

Here's a shove. Then, work your way toward the right side of the proof needed.

Use the Chain Rule for Partial Derivatives:

\(\displaystyle x\left(\frac{df}{ds}\frac{ds}{dx}+\frac{df}{dt}\frac{dt}{dx}\right)+y\left(\frac{df}{ds}\frac{ds}{dy}+\frac{df}{dt}\frac{dt}{dy}\right)\)

You can find \(\displaystyle \frac{ds}{dx}, \;\ \frac{ds}{dy}, \;\ \frac{dt}{dx}, \;\ \frac{dt}{dy}\) from your given s and t and sub those in.

Then, the only partials that remain are \(\displaystyle \frac{df}{ds}, \;\ \frac{df}{dt}\)

Do some factoring and it will fall into place.