Question Statement

Differentiate the following function with respect to $x$:

$y=x^2-\cos x$

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Background and Explanation

This problem requires applying the power rule for differentiation to the polynomial term $x^2$ and the standard trigonometric derivative for $\cos x$. Since the function is a difference of two terms, we differentiate each term separately and combine the results.

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Solution

We start with the given function:

$y=x^2-\cos x$

To find the derivative, we differentiate both sides with respect to $x$:

$\frac{dy}{dx}=\frac{d}{dx}\left(x^2\right)-\frac{d}{dx}(\cos x)$

Now we apply the appropriate differentiation rules to each term. For the first term, we use the power rule where $\frac{d}{dx}(x^n)=n{x}^{n-1}$, giving us $2x$. For the second term, we use the fact that the derivative of $\cos x$ is $-\sin x$:

$\frac{dy}{dx}=2x-(-\sin x)$

Simplifying the expression by resolving the double negative:

$\frac{dy}{dx}=2x+\sin x$

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Key Formulas or Methods Used

• Power Rule: $\frac{d}{dx}(x^n)=n{x}^{n-1}$
• Derivative of Cosine: $\frac{d}{dx}(\cos x)=-\sin x$
• Difference Rule: $\frac{d}{dx}[f(x)-g(x)]=f^{\prime }(x)-g^{\prime }(x)$

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Summary of Steps

1. Write down the given function: $y=x^2-\cos x$
2. Apply the differentiation operator to both sides with respect to $x$
3. Separate the derivative into individual terms using the difference rule: $\frac{d}{dx}(x^2)-\frac{d}{dx}(\cos x)$
4. Apply the power rule to differentiate $x^2$, yielding $2x$
5. Differentiate $\cos x$ to obtain $-\sin x$
6. Combine and simplify: $\frac{dy}{dx}=2x-(-\sin x)=2x+\sin x$