Partial Derivatives of xy ln(z^2 + 1)

Identify the constant factor first

Before differentiating, look at which factors involve the variable at hand. For $\frac{\partial f}{\partial x}$ the block $y\ln(z^2+1)$ contains no $x$, so the whole thing is one constant multiplier and the derivative of the remaining $x$ is $1$.

The $z$ partial is the only one that does real work. Now $xy$ is the constant and you differentiate $\ln(z^2+1)$ by the chain rule: one over the inside, times the derivative of the inside, which is $2z$.

The mistake: differentiating a constant block anyway

Seeing a logarithm makes many students differentiate it no matter which variable they were asked about, producing something like $\frac{y}{z^2+1}$ for the $x$ partial. The logarithm has no $x$ in it, so with respect to $x$ it is a number, and numbers ride along untouched.

The reverse mistake shows up in the $z$ partial, where students drop the $xy$ because those variables are held constant. Held constant means it stays as a multiplier. If you find yourself with an answer to a $z$ question that has no $x$ or $y$ in it, something has been deleted that should have survived.

• For the $x$ and $y$ partials, $\ln(z^2+1)$ is a constant factor and is copied down.
• For the $z$ partial, $xy$ is a constant factor and is copied down.
• Only one factor is ever differentiated here, because the variables never share a factor.

Checking the answers at a point

Take $(2,3,1)$. There $z^2+1 = 2$, so $\ln(z^2+1) = \ln 2 \approx 0.6931$. That gives $f_x = 3\ln 2 \approx 2.0794$ and $f_y = 2\ln 2 \approx 1.3863$, while $f_z = \frac{2(2)(3)(1)}{2} = 6$.

Two structural facts help confirm the shape of the answers. Multiplying the first two partial derivatives by their own variables both return $f$ itself, since $x f_x = y f_y = xy\ln(z^2+1)$. And on the plane $z = 0$ the logarithm is $\ln 1 = 0$, so all three partial derivatives vanish there, which is why the whole plane $z = 0$ consists of critical points.