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10. (a) Given that $y = (|x|^2 + 7)^{\frac{1}{2}}$, find $\frac{dy}{dx}$ - Scottish Highers Maths - Question 10 - 2016

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$10.-(a)-Given-that-$y-=-(|x|^2-+-7)^{\frac{1}{2}}$,-find-$\frac{dy}{dx}$-Scottish Highers Maths-Question 10-2016.png$

10. (a) Given that $y = (|x|^2 + 7)^{\frac{1}{2}}$, find $\frac{dy}{dx}$. (b) Hence find $\int \frac{4x}{\sqrt{x^2 + 7}} \, dx$.

Worked Solution & Example Answer:10. (a) Given that $y = (|x|^2 + 7)^{\frac{1}{2}}$, find $\frac{dy}{dx}$ - Scottish Highers Maths - Question 10 - 2016

Step 1

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Answer

We start by applying the chain rule to differentiate the function.

First, we differentiate $y$ with respect to $x$ :

$\frac{dy}{dx} = \frac{1}{2}(|x|^2 + 7)^{-\frac{1}{2}} \cdot \frac{d}{dx}(|x|^2 + 7).$

Now, we know that the derivative of $|x|^2$ is $2x$ , therefore:

$\frac{dy}{dx} = \frac{1}{2}(|x|^2 + 7)^{-\frac{1}{2}} \cdot 2x = \frac{x}{\sqrt{|x|^2 + 7}}.$

Step 2

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Answer

Using the result from part (a), we can now evaluate the integral.

We have:

$\int \frac{4x}{\sqrt{x^2 + 7}} \, dx = 4 \int \frac{x}{\sqrt{x^2 + 7}} \, dx.$

Substituting $y = \sqrt{x^2 + 7}$ gives:

$= 4 \cdot \frac{1}{2} (x^2 + 7)^{\frac{1}{2}} + C = 4 \sqrt{x^2 + 7} + C.$

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