Find the derivative of x^2e^(3xy)?

Question

find fx, fy. So i have the hang of the simpler stuff however im stuck on these types of questions. step by step would be really good :)

Answer 1

Derivative of (x^2)(e^3xy)

Derivative with respect to x

Than

d / dx [ (x^2)(e^3xy) ]

{ Note : d/dx [uv] = v d/dx(u) + u d/dx(v) }

(e^3xy)[d/dx(x^2)] + (x^2)[d/dx(e^3xy)]

Formula : d/dx(x^2) = 2x  ,  d/dx(e^ax) = a(e^ax)

(e^3xy)[ 2x ] + (x^2)3y(e^3xy)

2x(e^3xy) + 3y(x^2)(e^3xy)

There fore

d / dx [ (x^2)(e^3xy) ] = 2x(e^3xy) + 3y(x^2)(e^3xy)

 

 

Derivative of (x^2)(e^3xy)

Derivative with respect to y

Than

d/dy[(x^2)(e^3xy)]

{ Note : d/dx [uv] = v d/dx(u) + u d/dx(v) }

(e^3xy)[d/dy(x^2)] + (x^2)[d/dy(e^3xy)]

Formula : d/dy(constant) = 0  ,  d/dy(e^ay) = a(e^ay)

Here (x^2) is constant so d/dy(x^2) = 0

(e^3xy)(0) + (x^2)3x(e^3xy)

Simplify

0 + 3(x^2)(x)(e^3xy)

3(x^3)(e^3xy)                                (x^2)(x) = (x^3)

There fore

d/dy[(x^2)(e^3xy)] = 3(x^3)(e^3xy)

Answer 2

Differentiate with respect to x we get,

Differentiate with respect to x we get,