A point is moving along the graph of the given function such...

A point is moving along the graph of the given function such that $d x / d t$ is 2 centimeters per second. Find $d y / d t$ for the given values of $x .$
$y=\cos x \quad$ (a) $x=\frac{\pi}{6} \quad$ (b) $x=\frac{\pi}{4} \quad$ (c) $x=\frac{\pi}{3}$

Key Concepts

Chain Rule

The chain rule is a fundamental concept in calculus that allows you to differentiate composite functions. It states that if one variable depends on another which in turn depends on a third, then the derivative of the outer function with respect to the third variable is the product of the derivative of the outer function with respect to the middle variable and the derivative of the middle variable with respect to the third variable.

Parametric Differentiation

Parametric differentiation involves finding the derivative of one variable with respect to another when both are given as functions of a third parameter, typically time. By applying the chain rule, the rate of change of the dependent variable with respect to time is obtained as the product of the spatial derivative and the time derivative of the independent variable.

Differentiation of Trigonometric Functions

Differentiation of trigonometric functions refers to the process of computing derivatives of functions like sine and cosine. For instance, the derivative of cosine with respect to its variable is negative sine. This concept is essential when dealing with functions that involve trigonometric components.

Transcript

00:01 In problem 8, we have this problem dealing with related rates.

00:05 We have y and x, which are related.

00:07 They're both functions of time, and we're given the derivative of x with respect to t.

00:12 And we need to find the derivative of y with respect to t at these three different x values.

00:18 So in order to do this, we're going to find the derivative of this given function, and we're going to be using the chain rule.

00:25 So since this x is essentially x of t, it's a function of time.

00:30 That means that we have a function within a function.

00:33 So we have to take the derivative of cosine of x with respect to x, and then we're going to have to multiply that by the derivative of x with respect to t.

00:42 So let's begin.

00:44 First we take the derivative of y with respect to t, that's going to be equal to the derivative of cosine of x with respect to x, plus is going to be negative sign of x.

00:56 We're going to multiply that by the derivative of x with respect to t.

01:02 So we're just using the chain rule there.

01:04 Now that we have this, we can begin plugging in values.

01:08 So in part a, where x is equal to pi over 6, so we have the derivative of y with respect to t, it's going to be negative sine of pi over 6, and that's going to be multiplied by the derivative of x with respect to t, which we're given as 2 centimeter through second...

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