Question Statement

Differentiate $y=4x^3+x+\sin x$ with respect to $x$.

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Background and Explanation

This problem requires applying basic differentiation rules including the sum rule (differentiating term by term), the power rule for polynomial terms, and the standard derivative of trigonometric functions.

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Solution

We begin with the given function: $y=4x^3+x+\sin x$

To find $\frac{dy}{dx}$, we apply the sum rule for differentiation, which allows us to differentiate each term individually:

$\begin{array}{rl}\frac{dy}{dx}& =4\frac{d}{dx}\left(x^3\right)+\frac{d}{dx}(x)+\frac{d}{dx}(\sin x)\\ & =4\left(3x^2\right)+1+\cos x\\ \frac{dy}{dx}& =12x^2+1+\cos x\end{array}$

Therefore, the derivative is $\frac{dy}{dx}=12x^2+1+\cos x$.

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Key Formulas or Methods Used

• Sum Rule: $\frac{d}{dx}[f(x)+g(x)+h(x)]=\frac{d}{dx}f(x)+\frac{d}{dx}g(x)+\frac{d}{dx}h(x)$
• Power Rule: $\frac{d}{dx}(x^n)=n{x}^{n-1}$ (applied to the $x^3$ term)
• Derivative of $x$: $\frac{d}{dx}(x)=1$
• Derivative of sine: $\frac{d}{dx}(\sin x)=\cos x$
• Constant Multiple Rule: $\frac{d}{dx}[cf(x)]=c\frac{d}{dx}f(x)$ (applied to the coefficient $4$)

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Summary of Steps

1. Write down the given function: $y=4x^3+x+\sin x$
2. Apply the sum rule to separate the differentiation of each term
3. Differentiate $4x^3$ using the power rule and constant multiple rule: $4\cdot 3x^2=12x^2$
4. Differentiate $x$ to obtain $1$
5. Differentiate $\sin x$ to obtain $\cos x$
6. Combine all differentiated terms: $\frac{dy}{dx}=12x^2+1+\cos x$