Question Statement

Determine ${\mathbf{f}}^{\prime }(\mathbf{x})$ for the following functions:

(a) $f(x)=\sqrt{5}$

(b) $f(x)=\sqrt{5}x$

(c) $f(x)=5\sqrt{x}$

(d) $f(x)=\sqrt{5x}$

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Background and Explanation

This problem assesses your understanding of basic differentiation rules: the constant rule, power rule, constant multiple rule, and chain rule. Recall that radicals can be rewritten as fractional exponents ($\sqrt{x}=x^{1/2}$) to apply the power rule. Pay careful attention to whether $\sqrt{5}$ is being treated as a constant coefficient or if the variable $x$ appears inside the radical.

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Solution

Part (a)

The function $f(x)=\sqrt{5}$ is a constant function because $\sqrt{5}\approx 2.236$ is just a fixed number with no $x$ dependence.

Differentiating with respect to $x$:

$\begin{array}{rl}{f}^{\prime }(x)& =\frac{d}{dx}(\sqrt{5})\\ & =0\left(\because \frac{d}{dx}(c)=0\right)\end{array}$

Since the derivative of any constant $c$ is zero, $f^{\prime }(x)=0$.

Part (b)

The function $f(x)=\sqrt{5}x$ is a linear function where $\sqrt{5}$ acts as a constant coefficient multiplying the variable $x$.

Using the constant multiple rule, we factor out the constant and differentiate $x$:

$\begin{array}{rl}{f}^{\prime }(x)& =\sqrt{5}\cdot \frac{d}{dx}(x)\\ & =\sqrt{5}\cdot (1)\\ & =\sqrt{5}\end{array}$

Part (c)

For $f(x)=5\sqrt{x}$, we treat this as a constant $5$ multiplied by the function $\sqrt{x}=x^{1/2}$.

Apply the constant multiple rule, then use the power rule on $x^{1/2}$:

$\begin{array}{rl}{f}^{\prime }(x)& =5\frac{d}{dx}(\sqrt{x})\\ & =5\cdot \left(\frac{1}{2}{x}^{\frac{1}{2}-1}\right)\\ & =5\cdot \frac{1}{2}{x}^{-\frac{1}{2}}\\ & =\frac{5}{2x^{1/2}}\\ & =\frac{5}{2\sqrt{x}}\end{array}$

Note that $x^{-1/2}=\frac{1}{x^{1/2}}=\frac{1}{\sqrt{x}}$.

Part (d)

For $f(x)=\sqrt{5x}$, the variable $x$ appears inside the radical. We can view this as a composite function $(5x{)}^{1/2}$ and apply the chain rule, or alternatively rewrite it as $\sqrt{5}\sqrt{x}$. Here we use the chain rule approach:

$\begin{array}{rl}{f}^{\prime }(x)& =\frac{d}{dx}(\sqrt{5x})\\ & =\frac{1}{2}(5x{)}^{\frac{1}{2}-1}\cdot \frac{d}{dx}(5x)\\ & =\frac{1}{2}(5x{)}^{-\frac{1}{2}}\cdot 5\left(\frac{d}{dx}(x)\right)\\ & =\frac{1}{2}\frac{1}{(5x{)}^{1/2}}5(1)\\ & =\frac{5}{2\sqrt{5x}}\\ & =\frac{\sqrt{5}}{2\sqrt{x}}\end{array}$

The final simplification uses the fact that $\frac{5}{\sqrt{5}}=\sqrt{5}$.

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Key Formulas or Methods Used

• Constant Rule: $\frac{d}{dx}(c)=0$ for any constant $c$
• Power Rule: $\frac{d}{dx}(x^n)=n{x}^{n-1}$
• Constant Multiple Rule: $\frac{d}{dx}[c\cdot f(x)]=c\cdot f^{\prime }(x)$
• Chain Rule: $\frac{d}{dx}[f(g(x))]=f^{\prime }(g(x))\cdot g^{\prime }(x)$
• Exponent Conversion: $\sqrt{x}=x^{1/2}$ and $\sqrt[n]{x^m}=x^{m/n}$

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Summary of Steps

1. Identify the structure: Determine if the function is a constant, linear with constant coefficient, constant times power function, or composite function.
2. Apply the constant rule when $f(x)$ is a fixed number (part a).
3. Use the constant multiple rule to factor out coefficients like $\sqrt{5}$ or $5$ (parts b and c).
4. Convert radicals to exponents ($\sqrt{x}=x^{1/2}$) to apply the power rule, then subtract 1 from the exponent.
5. Apply the chain rule when the variable appears inside the radical (part d): differentiate the outer function (the power), then multiply by the derivative of the inner function ($5x$).
6. Simplify negative exponents by moving them to the denominator, and rationalize or reduce fractions where possible.