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Chapter 3: Problem 25

Find \(d y / d x\) for the following functions. $$y=\sin x \cos x$$

Short Answer

Expert verified
Answer: The derivative of the function is \(\frac{dy}{dx} = \cos^2 x - \sin^2 x\).

Step by step solution

01

Identify the functions u and v

In this problem, the functions u and v are: $$u(x) = \sin x$$ $$v(x) = \cos x$$ Now, we need to find the derivatives of these functions.
02

Find the derivatives of u(x) and v(x)

To find the derivative of $$u(x) = \sin x$$, differentiate using the basic rules of differentiation: $$u'(x) = \frac{d}{dx}[\sin x] = \cos x$$ Now, we need to find the derivative of $$v(x) = \cos x$$: $$v'(x) = \frac{d}{dx}[\cos x] = - \sin x$$
03

Apply the product rule

Now that we have the derivatives of u and v, we can apply the product rule to find the derivative of the given function. The product rule is: $$(u \cdot v)' = u' \cdot v + u \cdot v'.$$ So, for our function, we have: $$\frac{dy}{dx} = (\sin x \cdot \cos x)' = (\cos x)(\cos x) + (\sin x)(- \sin x)$$
04

Simplify the expression

Now, we'll simplify the expression for our derivative: $$\frac{dy}{dx} = \cos^2 x - \sin^2 x$$ So, the derivative of the given function is: $$\frac{dy}{dx} = \cos^2 x - \sin^2 x$$

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