Using identities Use the identity sin 2x=2 sin x cos x sin 2 to find d/dx (sin 2x). Then use the identity cos 2x = cos² x−sin² x to express the derivative of sin 2x in terms of cos 2x.

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Start by using the identity $ \sin 2x = 2 \sin x \cos x $. This identity helps us express $ \sin 2x $ in terms of $ \sin x $ and $ \cos x $.

Differentiate $ \sin 2x = 2 \sin x \cos x $ with respect to $ x $. Use the product rule for differentiation, which states that $ \frac{d}{dx}(u \cdot v) = u'v + uv' $. Here, $ u = \sin x $ and $ v = \cos x $.

Apply the product rule: $ \frac{d}{dx}(2 \sin x \cos x) = 2(\cos x \cdot \cos x + \sin x \cdot (-\sin x)) $. Simplify this expression to find $ \frac{d}{dx}(\sin 2x) $.

Now, use the identity $ \cos 2x = \cos^2 x - \sin^2 x $ to express $ \cos^2 x $ and $ \sin^2 x $ in terms of $ \cos 2x $. Substitute these into the derivative expression obtained in the previous step.

Simplify the expression to express $ \frac{d}{dx}(\sin 2x) $ in terms of $ \cos 2x $. This involves algebraic manipulation using the identity $ \cos 2x = \cos^2 x - \sin^2 x $.

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Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. They are essential for simplifying expressions and solving equations in calculus. The identities sin 2x = 2 sin x cos x and cos 2x = cos² x - sin² x are examples that help relate different trigonometric functions, making it easier to differentiate or integrate them.

Differentiation

Differentiation is a fundamental concept in calculus that involves finding the derivative of a function, which represents the rate of change of the function with respect to its variable. In this context, we apply the chain rule and product rule to differentiate sin 2x using the identity provided. Understanding how to differentiate trigonometric functions is crucial for solving problems related to motion, optimization, and other applications.

Chain Rule

The chain rule is a formula for computing the derivative of the composition of two or more functions. It states that if you have a function g(x) that is composed with another function f(x), the derivative is found by multiplying the derivative of f with respect to g by the derivative of g with respect to x. This rule is particularly useful when differentiating functions like sin(2x), where the inner function (2x) affects the outer function (sin).

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교과서 질문

Orthogonal trajectories Two curves are orthogonal to each other if their tangent lines are perpendicular at each point of intersection (recall that two lines are perpendicular to each other if their slopes are negative reciprocals). A family of curves forms orthogonal trajectories with another family of curves if each curve in one family is orthogonal to each curve in the other family. For example, the parabolas y = cx² form orthogonal trajectories with the family of ellipses x²+2y² = k, where c and k are constants (see figure).

Find dy/dx for each equation of the following pairs. Use the derivatives to explain why the families of curves form orthogonal trajectories. <IMAGE>

y = cx²; x²+2y² = k, where c and k are constants

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교과서 질문

Derivatives Find and simplify the derivative of the following functions.

f(x) = x /x+1

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교과서 질문

27–76. Calculate the derivative of the following functions.

$y = {(x^{2} + 2 x + 7)}^{8}$