Question

The directional derivative of $f(x,y) = x^2 + 3xy$ at the point (1,2) in the direction of the vector $3\hat{i} + 4\hat{j}$ is

• A. $\frac{22}{5}$
• B. $\frac{24}{5}$
• C. $\frac{12}{5}$
• D. $\frac{36}{5}$

Hint:

Always remember to convert the direction vector $\vec{v}$ into a unit vector $\vec{u}$ before calculating the dot product.
Using the non-unit vector directly is a common error that leads to incorrect values.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
This question asks for the directional derivative of a scalar function of two variables, $f(x, y)$, at a specified point in the direction of a given vector.

Step 2: Key Formula or Approach:
The directional derivative ($D_u f$) of a function $f$ in the direction of a unit vector $\vec{u}$ is given by the dot product of the gradient of the function and the unit vector:
\[ D_u f = \nabla f \cdot \vec{u} \]
where:
$\nabla f$ = gradient of the function $f$
$\vec{u}$ = unit vector in the direction of the given vector $\vec{v}$

Step 3: Detailed Explanation:

• The given function is:
\[ f(x, y) = x^2 + 3xy \]

• First, we find the gradient vector of $f(x, y)$:
\[ \nabla f(x, y) = \frac{\partial f}{\partial x} \hat{i} + \frac{\partial f}{\partial y} \hat{j} \]

• Calculate the partial derivatives:
\[ \frac{\partial f}{\partial x} = \frac{\partial}{\partial x} (x^2 + 3xy) = 2x + 3y \]
\[ \frac{\partial f}{\partial y} = \frac{\partial}{\partial y} (x^2 + 3xy) = 3x \]

• This gives the gradient vector as:
\[ \nabla f(x, y) = (2x + 3y)\hat{i} + (3x)\hat{j} \]

• Evaluate the gradient at the specified point $(1, 2)$:
\[ \nabla f(1, 2) = (2(1) + 3(2))\hat{i} + (3(1))\hat{j} = (2 + 6)\hat{i} + 3\hat{j} = 8\hat{i} + 3\hat{j} \]

• Next, find the unit vector $\vec{u}$ in the direction of the given vector $\vec{v} = 3\hat{i} + 4\hat{j}$.

• The magnitude of the vector $\vec{v}$ is:
\[ |\vec{v}| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

• The unit vector $\vec{u}$ is:
\[ \vec{u} = \frac{\vec{v}}{|\vec{v}|} = \frac{3\hat{i} + 4\hat{j}}{5} = \frac{3}{5}\hat{i} + \frac{4}{5}\hat{j} \]

• Finally, calculate the directional derivative by taking the dot product of $\nabla f(1, 2)$ and $\vec{u}$:
\[ D_u f = \nabla f(1, 2) \cdot \vec{u} = (8\hat{i} + 3\hat{j}) \cdot \left(\frac{3}{5}\hat{i} + \frac{4}{5}\hat{j}\right) \]
\[ D_u f = 8 \left(\frac{3}{5}\right) + 3 \left(\frac{4}{5}\right) = \frac{24}{5} + \frac{12}{5} = \frac{36}{5} \]

Step 4: Final Answer
The directional derivative is $\frac{36}{5}$, which matches option (D).