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Chapter 3: Problem 18

Find \(d y / d x\) for the following functions. $$y=5 x^{2}+\cos x$$

Short Answer

Expert verified
Answer: The derivative of the function is \(\frac{dy}{dx} = 10x - \sin x\).

Step by step solution

01

Identify the terms that need to be differentiated

The given function has two terms: 1. A polynomial term: \(5x^2\) 2. A trigonometric term: \(\cos x\) We will differentiate each term separately and then combine the results to find the derivative of the entire function.
02

Differentiate the polynomial term

To differentiate the polynomial term, we will use the power rule. The power rule states that if \(y = ax^n\), then \(\frac{dy}{dx} = nax^{n-1}\). Applying the power rule to our first term, we have: $$\frac{d}{dx}(5x^2) = 2(5)x^{2-1} = 10x$$ So, the derivative of the polynomial term, \(5x^2\), is \(10x\).
03

Differentiate the trigonometric term

To differentiate the trigonometric term, we will use the chain rule. The chain rule states that if \(y = f(g(x))\), then \(\frac{dy}{dx} = f'(g(x))g'(x)\). Applying the chain rule to \(\cos x\), we have: $$\frac{d}{dx}(\cos x) = -\sin x$$ So, the derivative of the trigonometric term, \(\cos x\), is \(-\sin x\).
04

Combine the derivatives

Now, we will combine the derivatives of the two terms found in steps 2 and 3 to find the derivative of the entire function: $$\frac{dy}{dx} = \frac{d}{dx}(5x^2 + \cos x) = \frac{d}{dx}(5x^2) + \frac{d}{dx}(\cos x) = 10x - \sin x$$ So, the derivative of the given function, \(y = 5x^2 + \cos x\), with respect to x is: $$\frac{dy}{dx} = 10x - \sin x$$

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