Question Statement

Find $\frac{dy}{dx}$ for the following functions:

(a) $y=\frac{x+2x^{3/2}}{\sqrt{x}}$

(b) $y=\left(x^3-5\right)(2x+3)$

(c) $y=\left(4x^2-3\right)\left(7x^2+x\right)$

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Background and Explanation

These problems require the power rule for differentiation ($\frac{d}{dx}{x}^n=n{x}^{n-1}$) combined with algebraic simplification skills. For parts (a), (b), and (c), we first simplify the expression (by dividing terms or expanding products) to make the differentiation process straightforward.

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Solution

Part (a)

For this rational expression, we simplify by dividing each term in the numerator by $\sqrt{x}$ before differentiating. This is easier than using the quotient rule.

First, rewrite $\sqrt{x}$ as $x^{1/2}$ and split the fraction:

$y=\frac{x}{x^{1/2}}+\frac{2x^{3/2}}{x^{1/2}}$

Apply the exponent rule $\frac{x^a}{x^b}=x^{a-b}$ to simplify each term:

$y=x^{1-\frac{1}{2}}+2x^{\frac{3}{2}-\frac{1}{2}}$

$y=x^{\frac{1}{2}}+2x^{\frac{2}{2}}$

$y=x^{\frac{1}{2}}+2x$

Now differentiate term by term using the power rule $\frac{d}{dx}(x^n)=n{x}^{n-1}$:

$\frac{dy}{dx}=\frac{d}{dx}\left(x^{1/2}\right)+2\frac{d}{dx}(x)$

$=\frac{1}{2}{x}^{\frac{1}{2}-1}+2(1)$

$=\frac{1}{2}{x}^{-\frac{1}{2}}+2$

Convert the negative exponent to a positive exponent in the denominator:

$=\frac{1}{2x^{1/2}}+2$

$=\frac{1}{2\sqrt{x}}+2$

Part (b)

Here we expand the product of two polynomials first, then differentiate term by term. (Alternatively, you could use the product rule, but expansion is straightforward here.)

Expand using the distributive property:

$y=x^3(2x)+x^3(3)-5(2x)-5(3)$

$y=2x^4+3x^3-10x-15$

Differentiate each term using the power rule. Remember that the derivative of a constant ($-15$) is zero:

$\frac{dy}{dx}=2\frac{d}{dx}\left(x^4\right)+3\frac{d}{dx}\left(x^3\right)-10\frac{d}{dx}(x)-\frac{d}{dx}(15)$

$=2\left(4x^3\right)+3\left(3x^2\right)-10(1)-0$

$\frac{dy}{dx}=8x^3+9x^2-10$

Part (c)

Similar to part (b), we first expand the product, then apply the power rule to each term.

Expand using the distributive property (FOIL method):

$y=4x^2(7x^2)+4x^2(x)-3(7x^2)-3(x)$

$y=28x^4+4x^3-21x^2-3x$

Differentiate term by term:

$\frac{dy}{dx}=28\frac{d}{dx}\left(x^4\right)+4\frac{d}{dx}\left(x^3\right)-21\frac{d}{dx}\left(x^2\right)-3\frac{d}{dx}(x)$

Apply the power rule to each term:

$=28\left(4x^3\right)+4\left(3x^2\right)-21(2x)-3(1)$

$=112x^3+12x^2-42x-3$

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Key Formulas or Methods Used

• Power Rule: $\frac{d}{dx}(x^n)=n{x}^{n-1}$
• Exponent Division: $\frac{x^a}{x^b}=x^{a-b}$ (used in part a)
• Negative Exponents: $x^{-n}=\frac{1}{x^n}$ or $x^{-1/2}=\frac{1}{\sqrt{x}}$
• Constant Rule: $\frac{d}{dx}(c)=0$ where $c$ is a constant
• Polynomial Expansion: Distributive property for multiplying binomials

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Summary of Steps

1. Simplify the expression: For rational functions (part a), divide terms to simplify exponents. For products of polynomials (parts b and c), expand into standard polynomial form.
2. Apply the power rule: Differentiate each term using $\frac{d}{dx}(x^n)=n{x}^{n-1}$.
3. Handle constants: Remember that the derivative of any constant is zero.
4. Simplify: Convert any negative exponents back to radical form or rational expressions if needed for the final answer.