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Given that $y = 12x^2 + 8oldsymbol{ ext{√}}x$, where $x > 0$, find \( \frac{dy}{dx} \. - Scottish Highers Maths - Question 2 - 2016

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Given-that-$y-=-12x^2-+-8oldsymbol{-ext{√}}x$,-where-$x->-0$,-find-\(-\frac{dy}{dx}-\.-Scottish Highers Maths-Question 2-2016.png

Given that $y = 12x^2 + 8oldsymbol{ ext{√}}x$, where $x > 0$, find \( \frac{dy}{dx} \.

Worked Solution & Example Answer:Given that $y = 12x^2 + 8oldsymbol{ ext{√}}x$, where $x > 0$, find \( \frac{dy}{dx} \. - Scottish Highers Maths - Question 2 - 2016

Step 1

Write in differentiable form

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Answer

The equation is already in a differentiable form as given:

y=12x2+8x12y = 12x^2 + 8x^{\frac{1}{2}}

Step 2

Differentiate first term

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Answer

Using the power rule, we differentiate the first term:

ddx(12x2)=24x\frac{d}{dx}(12x^2) = 24x

Step 3

Differentiate second term

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Answer

For the second term:

ddx(8x12)=8⋅12x−12=4x−12=4x\frac{d}{dx}(8x^{\frac{1}{2}}) = 8 \cdot \frac{1}{2}x^{-\frac{1}{2}} = 4x^{-\frac{1}{2}} = \frac{4}{\sqrt{x}}

Step 4

Combine results

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Answer

Now, we can combine the derivatives of both terms:

dydx=24x+4x\frac{dy}{dx} = 24x + \frac{4}{\sqrt{x}}

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