Question Statement

Find the derivative of $y=(7x+1)\left(x^4-x^3-9x\right)$ with respect to $x$.

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Background and Explanation

This problem requires the product rule for differentiation, which is essential when finding the derivative of two functions multiplied together. You should also be comfortable applying the power rule to differentiate polynomial terms of the form $x^n$.

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Solution

We are given the function: $y=(7x+1)\left(x^4-x^3-9x\right)$

This is a product of two functions, so we apply the product rule: $\frac{d}{dx}[u\cdot v]=u\frac{dv}{dx}+v\frac{du}{dx}$.

Let us identify:

• $u=7x+1$
• $v=x^4-x^3-9x$

First, compute the derivatives of each part: $\frac{du}{dx}=\frac{d}{dx}(7x+1)=7$

$\frac{dv}{dx}=\frac{d}{dx}\left(x^4-x^3-9x\right)=4x^3-3x^2-9$

Now apply the product rule formula:

$\frac{dy}{dx}=\left(x^4-x^3-9x\right)\frac{d}{dx}(7x+1)+(7x+1)\frac{d}{dx}\left(x^4-x^3-9x\right)$

Substitute the derivatives we found:

$=\left(x^4-x^3-9x\right)(7)+(7x+1)\left(4x^3-3x^2-9\right)$

Expand the first term: $=7x^4-7x^3-63x$

Expand the second term (distributing $7x$ and $1$): $+28x^4-21x^3-63x+4x^3-3x^2-9$

Combine all terms: $=7x^4-7x^3-63x+28x^4-21x^3-63x+4x^3-3x^2-9$

Group and simplify like terms:

• $x^4$ terms: $7x^4+28x^4=32x^4$
• $x^3$ terms: $-7x^3-21x^3+4x^3=-24x^3$
• $x^2$ terms: $-3x^2$
• $x$ terms: $-63x-63x=-126x$
• Constant: $-9$

Therefore: $\frac{dy}{dx}=32x^4-24x^3-3x^2-126x-9$

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Key Formulas or Methods Used

• Product Rule: $\frac{d}{dx}[u\cdot v]=u\frac{dv}{dx}+v\frac{du}{dx}$ (or equivalently $v\frac{du}{dx}+u\frac{dv}{dx}$)
• Power Rule: $\frac{d}{dx}[x^n]=n{x}^{n-1}$
• Sum/Difference Rule: $\frac{d}{dx}[f(x)\pm g(x)]=f^{\prime }(x)\pm g^{\prime }(x)$

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Summary of Steps

1. Identify the two functions being multiplied: $u=7x+1$ and $v=x^4-x^3-9x$
2. Differentiate each function separately: $\frac{du}{dx}=7$ and $\frac{dv}{dx}=4x^3-3x^2-9$
3. Apply the product rule formula: $\frac{dy}{dx}=v\frac{du}{dx}+u\frac{dv}{dx}$
4. Expand all terms by distributing the coefficients through the parentheses
5. Combine like terms (group by powers of $x$) to obtain the final simplified polynomial derivative