Finding the Second Derivative of a Function with Two Variables

In summary, the quotient rule states that z= f/g, where z' is the derivative of z with respect to x, and f' and g' are the derivatives of f and g, respectively. In order to find the second derivative, you need to find the derivative of z with respect to y, and then use the same principles to find the second derivative of y with respect to x.
  • #1
Maniac_XOX
86
5
Homework Statement
find first and second partial derivative of z= tan (x^2*y^2)
Relevant Equations
tan (x^2*y^2)= sin(x^2*y^2)/cos(x^2*y^2)
Quotient rule: z= f/g ------ z'= (f'g - g'f)/g^2
starting with finding the derivative in respect to x, i treated y^2 as constant 'a': z'= [(a*2x*cos a*x^2)(sin a*x^2) - (- a*2x*sin ax^2)]/cos(a*x^2)^2=
[(a*2x*cos a*x^2)(sin a*x^2)+(a*2x*sin ax^2)]/cos(a*x^2)^2
For the derivative in respect to y it would be the same process, how do i go from here to finding the second derivative?
 
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  • #2
Maniac_XOX said:
Quotient rule: z= f/g ------ z'= (f'g - g'f)/g^2
starting with finding the derivative in respect to x, i treated y^2 as constant 'a': z'= [(a*2x*cos a*x^2)(sin a*x^2) - (- a*2x*sin ax^2)]/cos(a*x^2)^2=
[(a*2x*cos a*x^2)(sin a*x^2)+(a*2x*sin ax^2)]/cos(a*x^2)^2
For the derivative in respect to y it would be the same process, how do i go from here to finding the second derivative?

Well, you take the first derivative, and you have a new function of ##x## and ##y##. What's the difficulty in taking a derivative of that? The second derivative uses the same principles as the first derivative.
 
  • #3
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In your case, I think you're not looking for the mixed partia derivatives, so go with the 1st and 4th formulas.
Source : https://www.khanacademy.org/math/mu...radient-articles/a/second-partial-derivatives
 
  • #5
  • #6
stevendaryl said:
Well, you take the first derivative, and you have a new function of ##x## and ##y##. What's the difficulty in taking a derivative of that? The second derivative uses the same principles as the first derivative.
You're right i was just unsure of the result I would get cos there is a lot of working out and wanted to be sure I made no arithmetic errors. I guess I just need to post the final answer and ask wether there are arithmetic errors. Thank you for your opinon x
 

1. What is calculus with two variables?

Calculus with two variables is a branch of mathematics that deals with the study of functions of two variables. It involves the use of mathematical concepts such as limits, derivatives, and integrals to analyze and solve problems involving two variables.

2. How is calculus with two variables different from single variable calculus?

In single variable calculus, we study functions of a single variable, while in calculus with two variables, we study functions of two variables. This means that in two variable calculus, we have to consider how the function changes in two different directions, instead of just one.

3. What are some real-world applications of calculus with two variables?

Calculus with two variables has many real-world applications, such as in physics, engineering, economics, and biology. It is used to model and analyze systems that involve two changing quantities, such as the velocity and acceleration of an object, or the supply and demand of a product.

4. What are some common techniques used in solving problems in calculus with two variables?

Some common techniques used in solving problems in calculus with two variables include finding partial derivatives, using the chain rule, and setting up and solving systems of equations. It is also important to understand the geometric interpretation of functions in two variables.

5. How can I improve my understanding of calculus with two variables?

To improve your understanding of calculus with two variables, it is important to practice solving problems and to have a solid understanding of single variable calculus. It can also be helpful to visualize functions in two variables and to seek out additional resources, such as textbooks, online tutorials, or working with a tutor.

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