I) Find the directional derivative of 𝑓(𝑥, 𝑦) = 𝑥 sin 𝑦 at (1,0) in the direction of the unit vector that make an angle of 𝜋/4 with positive 𝑥-axis.



Answer to a math question I) Find the directional derivative of 𝑓(𝑥, 𝑦) = 𝑥 sin 𝑦 at (1,0) in the direction of the unit vector that make an angle of 𝜋/4 with positive 𝑥-axis.

Expert avatar
56 Answers
To find the directional derivative of a function, we need to calculate the dot product of the gradient vector of the function and the unit vector in the given direction.

The gradient vector of the function f(x, y) is given by the partial derivatives of f with respect to each variable.

Let's find the partial derivatives of f(x, y):

\frac{\partial f}{\partial x} = \frac{\partial}{\partial x} (x \sin y) = \sin y

\frac{\partial f}{\partial y} = \frac{\partial}{\partial y} (x \sin y) = x \cos y

Now, we can find the gradient vector:

\nabla f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}\right) = \left(\sin y, x \cos y\right)

The unit vector that makes an angle of 𝜋/4 with the positive 𝑥-axis is:


Now, we can calculate the dot product of the gradient vector and the unit vector:

\nabla f\cdot\mathbf{u}=\left(\sin y,x\cos y\right)\cdot\left(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\right)=\frac{\sqrt{2}}{2}\sin y+\frac{\sqrt{2}}{2}x\cos y

Substituting the coordinates of the point (1, 0) into the equation above:

\nabla f\cdot\mathbf{u}=\frac{\sqrt{2}}{2}\cdot\sin0+\frac{\sqrt{2}}{2}\cdot1\cdot\cos0=\frac{\sqrt{2}}{2}\cdot0+\frac{\sqrt{2}}{2}\cdot1\cdot1=\frac{\sqrt{2}}{2}

Answer: The directional derivative of f(x, y) = x sin y at (1, 0) in the direction of the unit vector that makes an angle of 𝜋/4 with the positive 𝑥-axis is \frac{\sqrt{2}}{2} .

Frequently asked questions (FAQs)
Question: Find the domain and range of the cube root function f(x) = ∛x.
Question: What is the equation of the parabola with its graph opening downwards, vertex at (2, -5), and passing through the point (1, -6)?
Question: In a circle with radius 10 cm, if an angle at the center measures 60°, what is the length of the arc it intercepts?
New questions in Mathematics
How much volume of water in MegaLiters (ML) is required to irrigate 30 Hectare crop area with depth of 20mm?
If L = (-2, -5) is reflected across y= -4 , what are the coordinates of L?
A, B, C and D are numbers; If ABCD = 23, What is the result of ABCD BCDA CDAB DABC operation?
1 plus 1
B - (-4)=10
In a store there are packets of chocolate, strawberry, tutti-frutti, lemon, grape and banana sweets. If a person needs to choose 4 flavors of candy from those available, how many ways can they make that choice?
Suppose SAT reading scores are normally distributed with a mean of 496 and a standard deviation of 109. The University plans towards scholarships for students who scores are in the top 7%. What is the minimum score required for the scholarship round your answer to the nearest whole number.
Divide 22 by 5 solve it by array and an area model
logy/logx + logz/logy + logt/logz = 8x².t x=?
Suppose the Golf ball market is perfectly competitive and the functions are known: Q = 120 – 2Px – 2Py 0.2I Q = 2Px 40 Where I = Consumers' income ($200) and Py = Price of Good Y (40) Calculate the equilibrium elasticity: a) 1.6 b) -6 c) 6 d) 0.6
Two business partners have a bank balance of $17,942.00. After the first year their interest brings their balance to $18,928.91. What rate of interest is earned?
What’s the slope of a tangent line at x=1 for f(x)=x2. We can find the slopes of a sequence of secant lines that get closer and closer to the tangent line. What we are working towards is the process of finding a “limit” which is a foundational topic of calculus.
In a laboratory test, it was found that a certain culture of bacteria develops in a favorable environment, doubling its population every 2 hours. The test started with a population of 100 bacteria. After six hours, it is estimated that the number of bacteria will be:
Congratulations, you have saved well and are ready to begin your retirement. If you have $1,750,000.00 saved for your retirement and want it to last for 40 years, and will earn 10.8% compounded monthly: What is the amount of the monthly distribuion? 216.50 How much interest is earned in retirement?
When Sara was 15 years old, an uncle left her as inheritanceà a sum of 10,000 euros which he invested in a bank that applies the interest rate of 2,5% annual. Today Sara is 18 years and wants to buy a'car, how much she can ò withdraw from the bank?
The grading on a $159,775 house comes to $3974.75. What percent of the total cost is this? (Express your answer to the nearest hundredth percent.)
Derivative of 2x
Carmen's age was twice as old as Luis was when Carmen was Luis's age. When Luis is Carmen's age, their ages will add up to 112.
f(r) = 1/r+9 find f(x^2) + 1
In an experiment to assess the effect of listening to audiobooks while driving, participants were asked to drive down a straight road in a driving simulator. The accompanying data on time (in milliseconds) to react when a pedestrian walked into the street for 10 drivers listening to an audiobook are consistent with summary statistics and graphs that appeared in the paper "Good Distractions: Testing the Effect of Listening to an Audiobook on Driving Performance in Simple and Complex Road Environments."† (Round your answers to four decimal places.) 1,018 1,007 1,054 988 937 1,030 1,065 1,011 860 1,106 A button hyperlink to the SALT program that reads: Use SALT. Calculate the variance for this data set. 7437.7333 Incorrect: Your answer is incorrect. Calculate the standard deviation for this data set. 86.2022 Incorrect: Your answer is incorrect.