(a) If V is the volume of a cube with edge length x and the...

(a) If $V$ is the volume of a cube with edge length $x$ and the cube expands as time passes, find $d V / d t$ in terms of $d x / d t$ (b) If the length of the edge of a cube is increasing at a rate of $4 \mathrm{~cm} / \mathrm{s}$, how fast is the volume of the cube increasing when the edge length is $15 \mathrm{~cm} ?$

(a) If $V$ is the volume of a cube with edge length $x$ and the cube expands as time passes, find $d V / d t$ in terms of $d x / d t$
(b) If the length of the edge of a cube is increasing at a rate of $4 \mathrm{~cm} / \mathrm{s}$, how fast is the volume of the cube increasing when the edge length is $15 \mathrm{~cm} ?$

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Key Concepts

Chain Rule

The chain rule is a fundamental technique in calculus used to differentiate a composite function, where one variable depends on another which in turn depends on time. This rule allows us to compute the derivative of the outer function with respect to the inner variable and then multiply by the derivative of the inner variable with respect to time, enabling the evaluation of how changes in one quantity affect another over time.

Related Rates

Related rates problems involve finding the rate at which one quantity changes by relating it to another quantity that is changing with time. The approach requires differentiating a relation between the quantities with respect to time, then substituting known rates and values to solve for the unknown rate, making it a powerful tool in dynamic systems analysis.

Differentiation of Power Functions

Differentiation of power functions pertains to computing the derivative of functions of the form f(x) = x?, a basic operation in calculus. This concept is crucial when dealing with formulas like the volume of a cube (which is a power function of its edge length), as it allows for the determination of how small changes in one variable can affect the function as a whole.

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