Find the first partial derivatives of the function.
f(x, y) = x6 + 6xy7
fx(x, y)
=
Correct: Your answer is correct.
fy(x, y)
=

Given the function \( f(x, y) = x^6 + 6xy^7 \), we can now find the first partial derivatives with respect to both \(x\) and \(y\).

### 1. Partial derivative with respect to \(x\):
The derivative of \( f(x, y) = x^6 + 6xy^7 \) with respect to \(x\) is:
\[
f_x(x, y) = \frac{\partial}{\partial x} \left( x^6 + 6xy^7 \right)
\]
\[
f_x(x, y) = 6x^5 + 6y^7
\]

### 2. Partial derivative with respect to \(y\):
Now, let's differentiate \( f(x, y) = x^6 + 6xy^7 \) with respect to \(y\):
\[
f_y(x, y) = \frac{\partial}{\partial y} \left( x^6 + 6xy^7 \right)
\]
Since \(x^6\) is constant with respect to \(y\), its derivative is 0. So we only differentiate the second term:
\[
f_y(x, y) = 6x \cdot 7y^6 = 42xy^6
\]

### Final answers:
\[
f_x(x, y) = 6x^5 + 6y^7
\]
\[
f_y(x, y) = 42xy^6
\]

Would you like more details or have any questions?

Here are some related questions you can explore:
1. How do second partial derivatives differ from first partial derivatives?
2. What is the significance of mixed partial derivatives?
3. How would the partial derivatives change if the function had more variables?
4. What happens when you evaluate the partial derivatives at specific points?
5. How can partial derivatives be used in optimization problems?

**Tip:** Always remember to treat other variables as constants when calculating partial derivatives with respect to one variable!

Find the first partial derivatives of the function f(x, y) = x^6 + 6xy^7.

Learn how to calculate the first partial derivatives of the function f(x, y) = x^6 + 6xy^7. This solution covers multivariable calculus concepts, including the partial derivative rule and the product rule. Suitable for advanced high school or university-level students.

Math Problem Statement

Find the first partial derivatives of the function. f(x, y) = x6 + 6xy7 fx(x, y)

Correct: Your answer is correct. fy(x, y)

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation