Question Statement

Find the derivative of the following:

(a) $y=x^9$

(b) $f(x)=4x^{1/3}$

(c) $f(x)=9$

(d) $f(x)=6x^3+3x^2-10$

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Background and Explanation

This problem requires applying basic differentiation rules, specifically the Power Rule for terms with variables and the Constant Rule for constant terms. You will also use the Sum Rule to differentiate polynomials term by term.

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Solution

Part (a): $y=x^9$

We start with the function: $y=x^9$

To find the derivative, we differentiate with respect to $x$ using the Power Rule $\frac{d}{dx}(x^n)=n{x}^{n-1}$:

$\begin{array}{rl}\frac{dy}{dx}& =\frac{d}{dx}\left(x^9\right)\\ & =9x^{9-1}\because \frac{d}{dx}\left(x^n\right)=n{x}^{n-1}\\ & =9x^8\end{array}$

Result: $\frac{dy}{dx}=9x^8$

Part (b): $f(x)=4x^{1/3}$

We start with the function: $f(x)=4x^{1/3}$

Differentiate with respect to $x$. We apply the Constant Multiple Rule (pull out the 4) and then the Power Rule:

$\begin{array}{rl}{f}^{\prime }(x)& =4\cdot \frac{d}{dx}\left(x^{1/3}\right)\\ & =4\cdot \left(\frac{1}{3}{x}^{\frac{1}{3}-1}\right)\\ & =\frac{4}{3}{x}^{-2/3}\end{array}$

We can rewrite the negative exponent as a fraction in the denominator: $f^{\prime }(x)=\frac{4}{3x^{2/3}}$

Result: $f^{\prime }(x)=\frac{4}{3}{x}^{-2/3}$ or $\frac{4}{3x^{2/3}}$

Part (c): $f(x)=9$

We start with the constant function: $f(x)=9$

Differentiate with respect to $x$. The derivative of any constant is zero:

$\begin{array}{rl}{f}^{\prime }(x)& =\frac{d}{dx}(9)\\ & =0\left(\because \frac{d}{dx}(c)=0\right)\end{array}$

Result: $f^{\prime }(x)=0$

Part (d): $f(x)=6x^3+3x^2-10$

We start with the polynomial: $f(x)=6x^3+3x^2-10$

Differentiate with respect to $x$ using the Sum Rule (differentiate each term separately) and the Power Rule:

$\begin{array}{rl}{f}^{\prime }(x)& =6\frac{d}{dx}\left(x^3\right)+3\frac{d}{dx}\left(x^2\right)-\frac{d}{dx}(10)\\ & =6(3x^2)+3(2x)-0\\ & =18x^2+6x\end{array}$

Result: $f^{\prime }(x)=18x^2+6x$

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Key Formulas or Methods Used

• Power Rule: $\frac{d}{dx}(x^n)=n{x}^{n-1}$ for any real number $n$
• Constant Multiple Rule: $\frac{d}{dx}[c\cdot f(x)]=c\cdot f^{\prime }(x)$
• Constant Rule: $\frac{d}{dx}(c)=0$ where $c$ is any constant
• 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 function type (power function, constant, or polynomial)
2. Apply the Power Rule $\frac{d}{dx}(x^n)=n{x}^{n-1}$ to each variable term
3. Apply the Constant Multiple Rule to keep coefficients separate from the differentiation
4. Apply the Constant Rule (derivative equals 0) to any standalone constants
5. Combine results using the Sum/Difference Rule for polynomials
6. Simplify the final expression (convert negative exponents to fractions if needed)