For the demand equations find the rate of change of p with...

For the demand equations find the rate of change of p with respect to $x$ by differentiating implicitly $(x$ is the number of items that can be sold at a price of $\$ p$ ).
$$
x=\sqrt{10,000-p^{2}}
$$

Key Concepts

Implicit Differentiation

Implicit differentiation is a technique used when a function is defined implicitly by an equation rather than explicitly. It allows us to differentiate both sides of an equation with respect to a given variable, treating the other variable as an implicit function of the one we differentiate with respect to. This method is particularly useful when it is difficult or impossible to solve for one variable in terms of the other.

Chain Rule

The chain rule is a fundamental principle used in calculus to compute the derivative of a composite function. When an expression involves a function inside another function, the derivative is found by taking the derivative of the outer function with respect to the inner function and then multiplying it by the derivative of the inner function. This rule is essential in many differentiation problems, including when differentiating expressions involving nested functions like square roots.

Differentiation of Square Root Functions

Differentiation of square root functions involves applying standard differentiation rules along with the chain rule. The derivative of a square root function, expressed as sqrt(u), is calculated by differentiating the outer square root function to obtain 1/(2sqrt(u)) and then multiplying by the derivative of the inner function u. This approach is commonly used when dealing with functions that involve square roots of expressions dependent on another variable.

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