Questions

$$\frac{\partial}{\partial x}\big(xe^{yz}\big)=\ ?$$

$$\frac{\partial}{\partial y}\big(xe^{yz}\big)=\ ?$$

$$\frac{\partial}{\partial x}\,\ln(x^2+y^2)=\ ?$$

$$\frac{\partial}{\partial y}\,\big(x^2y^3+\sin(xy)\big)=\ ?$$

$$\frac{\partial^2}{\partial x\,\partial y}\,e^{xy}=\ ?$$

$$\frac{\partial}{\partial x}\,(x+yz)^{3}=\ ?$$

$$\frac{\partial}{\partial x}\,\big(y^{2}\cos(xy)\big)=\ ?$$

$$\frac{\partial}{\partial x}\,\frac{xyz}{x+y+z}=\ ?$$

9. Evaluate at the point

$$f(x,y)=\sqrt{x^{2}+y^{2}},\qquad \frac{\partial f}{\partial y}(3,4)=\ ?$$

Solutions

1)

Partial derivatives are not as hard as they look!

Basically, just differentiate with respect to the given variable and treat the other variables as constants!

For example,

$$\frac{\partial}{\partial x} (x^2 + 8y) = 2x$$ $$\frac{\partial}{\partial y} (x^2 + 8y) = 8$$

Therefore,

$$\frac{\partial}{\partial x} (xe^{yz}) = e^{yz}$$

2)
Treat $x,z$ as constants; chain rule on $e^{yz}$:

$$\frac{\partial}{\partial y}\big(xe^{yz}\big) = x\cdot e^{yz}\cdot z = xz\,e^{yz}.$$

3)

$$\frac{\partial}{\partial x}\ln(x^2+y^2) =\frac{1}{x^2+y^2}\cdot 2x =\frac{2x}{x^2+y^2}.$$

4)

$$\frac{\partial}{\partial y}\big(x^2y^3+\sin(xy)\big) = x^2\cdot 3y^2 + \cos(xy)\cdot x = 3x^2y^2 + x\cos(xy).$$

5)
First $\partial/\partial y$: $\frac{\partial}{\partial y}e^{xy}=x e^{xy}$.
Then $\partial/\partial x$:

$$\frac{\partial}{\partial x}\big(x e^{xy}\big) = e^{xy}+xy\,e^{xy} = e^{xy}(1+xy).$$

6)
Hold $y,z$ fixed:

$$\frac{\partial}{\partial x}(x+yz)^3 =3(x+yz)^2.$$

7)
Product/chain with $y$ treated constant:

$$\frac{\partial}{\partial x}\big(y^2\cos(xy)\big) = y^2\big(-\sin(xy)\big)\cdot y = -y^{3}\sin(xy).$$

8)
Quotient; $N=xyz,\ D=x+y+z$.
$N_x=yz,\ D_x=1$.

$$\frac{\partial}{\partial x}\frac{xyz}{x+y+z} =\frac{(yz)(x+y+z)-xyz\cdot 1}{(x+y+z)^2} =\frac{yz(y+z)}{(x+y+z)^2}.$$

9)

$$f_y=\frac{y}{\sqrt{x^2+y^2}}\;\Rightarrow\; f_y(3,4)=\frac{4}{\sqrt{3^2+4^2}}=\frac{4}{5}.$$