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Chapter 6: Problem 35

If \(y=e^{x} \cos x\), prove that \(\frac{d y}{d x}=\sqrt{2} e^{x} \cos \left(x+\frac{\pi}{4}\right)\).

Short Answer

Expert verified
To verify the given derivative, first differentiate the given expression using the product rule. The resulting derivative involves \(\cos(x)\) and \(-\sin(x)\) terms. Then we manipulate this to match the given result, using the cosine of sum formula. The final derivative is, in fact, \(\frac{dy}{dx}= \sqrt{2}e^{x}\cos\Big(x+\frac{\pi}{4}\Big)\).

Step by step solution

01

Apply the Product Rule

Given \(y=e^{x} \cos x\), apply the product rule which states: the derivative of a product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function. Thus, the derivative would be \(\frac{dy}{dx} = \frac{d}{dx}(e^{x})\cos(x) + e^{x}\frac{d}{dx}(\cos(x))\).
02

Differentiate Individual Terms

Differentiating \(e^{x}\) with respect to \(x\), we get \(e^{x}\), and differentiating \(\cos(x)\) with respect to \(x\), we get \(-\sin(x)\). Substituting these into the equation from Step 1, we get \(\frac{dy}{dx} = e^{x}\cos(x) - e^{x}\sin(x)\).
03

Identify the Trigonometric Identity to be used

Next, in order to get the required result, we need to identify a relevant trigonometric identity. The given result structure suggests using the cosine of sum formula, \(\cos(a+b)=\cos(a)\cos(b) - \sin(a)\sin(b)\). Therefore, we need our result to be expressed in the form \(\cos(x+\phi)\), which indicates that the sum of two functions is used, each multiplied with corresponding cosine and sine.
04

Apply the Trigonometric Identity

Expressing \(\frac{dy}{dx}\) in the form \(\cos(x+\phi)\) and using identity for cosine of sum we get \(\frac{dy}{dx} = \sqrt{2}e^{x}\big(\frac{\cos(x)}{\sqrt{2}} - \frac{\sin(x)}{\sqrt{2}}\big) = \sqrt{2}e^{x}\cos\Big(x+\frac{\pi}{4}\Big)\).

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