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A curve $C$ has equation $\newline$ $y=2+10x^{\frac{1}{2}}-2x^{\frac{3}{2}},\quad x > 0$ $\newline$ $(a)$ Find $\frac{dy}{dx}$ giving your answer in simplest form.

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Write a quadratic function from its x-intercepts and another point

Full solution

Q. A curve

C

has equation

\newline

y=2+10x^{\frac{1}{2}}-2x^{\frac{3}{2}},\quad x > 0

\newline

(a)

Find

\frac{dy}{dx}

giving your answer in simplest form.

Identify Derivatives: Differentiate each term of $y = 2 + 10x^{1/2} - 2x^{3/2}$ with respect to $x$ . For the constant term $2$ , the derivative is $0$ . For the term $10x^{1/2}$ , use the power rule: $\frac{d}{dx}[x^n] = n\cdot x^{n-1}$ . So, the derivative is $10\cdot(1/2)\cdot x^{(1/2)-1} = 5x^{-1/2}$ . For the term $-2x^{3/2}$ , again use the power rule: the derivative is $-2\cdot(3/2)\cdot x^{(3/2)-1} = -3x^{1/2}$ .
Apply Power Rule: Combine the derivatives of the individual terms to get the derivative of $y$ . $(\frac{dy}{dx}) = 0 + 5x^{(-\frac{1}{2})} - 3x^{(\frac{1}{2})}$ .
Combine Derivatives: Simplify the expression for $(\frac{dy}{dx})$ . $(\frac{dy}{dx}) = \frac{5}{x^{\frac{1}{2}}} - 3x^{\frac{1}{2}}.$

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