Determine the components of the gradient of f(x, y, z) = 3x⁴yz³ in the directions of the Cartesian unit vectors i, j and k.

Step-by-step solution

1. 1

   Recognise what "component along $\mathbf{i}$" means. For a unit vector $\mathbf{u}$ the component of $\nabla f$ along $\mathbf{u}$ is $\nabla f\cdot\mathbf{u}$, which is also the directional derivative $D_{\mathbf{u}}f$. Because $\mathbf{i},\mathbf{j},\mathbf{k}$ are the coordinate axes themselves, those three dot products are simply the three partial derivatives:

   $\nabla f=\frac{\partial f}{\partial x}\mathbf{i}+\frac{\partial f}{\partial y}\mathbf{j}+\frac{\partial f}{\partial z}\mathbf{k}.$

   So no dot products need to be computed — the question reduces to three one-variable derivatives.

2. 2

   Differentiate with respect to $x$, freezing $y$ and $z$. The factor $3yz^{3}$ is a constant here, so only $x^{4}$ is differentiated:

   $\frac{\partial f}{\partial x}=3yz^{3}\cdot 4x^{3}=12x^{3}yz^{3}.$

3. 3

   Differentiate with respect to $y$. Now $3x^{4}z^{3}$ is the constant and $y$ appears to the first power, so its derivative is $1$:

   $\frac{\partial f}{\partial y}=3x^{4}z^{3}.$

   Notice the $y$ disappears entirely — a useful sanity check, since $f$ was linear in $y$.

4. 4

   Differentiate with respect to $z$. Holding $3x^{4}y$ fixed and applying the power rule to $z^{3}$:

   $\frac{\partial f}{\partial z}=3x^{4}y\cdot 3z^{2}=9x^{4}yz^{2}.$

5. 5

   Assemble the gradient and verify by a consistency test. Collecting the three results,

   $\nabla f=\langle 12x^{3}yz^{3},\;3x^{4}z^{3},\;9x^{4}yz^{2}\rangle.$

   Each component keeps total degree $7$, one less in exactly the variable differentiated, exactly as expected for a monomial of degree $8$. Equivalently, $x f_x + y f_y + z f_z = (4+1+3)f = 8f$, Euler's identity for a homogeneous function of degree $8$.