Question

Find both first partial derivatives. $$ f(x, y)=x^{2}=3 y^{2}+7 $$

   Find both first partial derivatives.
$$
f(x, y)=x^{2}=3 y^{2}+7
$$

Calculus

Ron Larson, Robert… 8th Edition

Chapter 13, Problem 6

Instant Answer

Solved by Expert William Semus

09/25/2021

Step 1

This means we treat $y$ as a constant and differentiate the function with respect to $x$. The function is $f(x, y)=x^{2}-3y^{2}+7$. The derivative of $x^{2}$ with respect to $x$ is $2x$. The derivative of $-3y^{2}$ with respect to $x$ is $0$ because we are Show more…

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Find both first partial derivatives. $$ f(x, y)=x^{2}=3 y^{2}+7 $$

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Key Concepts

Chain Rule

The chain rule is a fundamental concept in calculus used to differentiate composite functions. When applying implicit differentiation, the chain rule is essential for correctly handling the differentiation of terms where one variable is a function of another, by multiplying the derivative of the outer function by the derivative of the inner function.

Implicit Differentiation

This is a method used to differentiate equations where the dependent variable is not isolated; the relationship between variables is given in an implicit form rather than as an explicit function. It involves differentiating both sides of the equation with respect to a particular variable while treating other variables as functions of that variable, allowing for the calculation of derivatives even when the function is not explicitly solved.

Partial Derivatives

Partial derivatives represent the rate at which a multivariable function changes with respect to one variable, holding the other variables constant. They are crucial in multivariable calculus for analyzing the behavior of functions in several dimensions, and are used in areas such as optimization, physics, and engineering to understand how changes in one input affect the output.

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