A particle moves along the x -axis so that its position at any time t ≥0 is given by x(t) . Find

step 1 For this problem, we are told that a particle moves along the x -axis so that its position at an

A particle moves along the x -axis so that its position at...

A particle moves along the $x$ -axis so that its position at any time $t \geq 0$ is given by $x(t) .$ Find the velocity at the indicated value of $t.$
$$x(t)=\sin ^{-1}\left(\frac{\sqrt{t}}{4}\right), \quad t=4$$

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Key Concepts

Velocity

Velocity is defined as the instantaneous rate of change of a particle's position with respect to time, which in mathematical terms is the derivative of the position function. It quantifies how fast and in what direction the position is changing, and its calculation is fundamental in kinematics and dynamics.

Chain Rule

The chain rule is a method used in differentiation to compute the derivative of composite functions. When a function is composed of an outer function and an inner function, the derivative is found by differentiating the outer function with respect to the inner function and then multiplying by the derivative of the inner function. This rule is essential when dealing with functions where one variable depends on another through an intermediate relationship.

Differentiation of Inverse Trigonometric Functions

Differentiation of inverse trigonometric functions involves applying specific formulas that arise from their relationship with standard trigonometric functions. For example, the derivative of the arcsine function is given by 1/sqrt(1 - u^2), where u is a function of the variable in question. This technique is critical when handling problems involving inverse trigonometric functions, as it allows for the determination of rates of change even in more complex nested or composite cases.

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