A-Level Edexcel Maths Pure Graphs of Functions: Given that $y = 3x^2 + 4 ext{sqr

Given that $y = 3x^2 + 4 ext{sqrt}(x), x > 0$, find (a) $\frac{dy}{dx}$ - Edexcel - A-Level Maths Pure - Question 5 - 2007 - Paper 1

Question 5

$Given-that-$y-=-3x^2-+-4-ext{sqrt}(x),-x->-0$,-find--(a)-$\frac{dy}{dx}$-Edexcel-A-Level Maths Pure-Question 5-2007-Paper 1.png$

Given that $y = 3x^2 + 4 ext{sqrt}(x), x > 0$, find (a) $\frac{dy}{dx}$. (b) $\frac{d^2y}{dx^2}$. (c) $\int y \, dx$.

Worked Solution & Example Answer:Given that $y = 3x^2 + 4 ext{sqrt}(x), x > 0$, find (a) $\frac{dy}{dx}$ - Edexcel - A-Level Maths Pure - Question 5 - 2007 - Paper 1

Step 1

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Answer

To find the first derivative of $y$ with respect to $x$ , we differentiate each term of the function:

For $3x^2$ , the derivative is $6x$ .
For $4\sqrt{x}$ , we can rewrite it as $4x^{1/2}$ , which gives us $\frac{4}{2}x^{-1/2} = 2x^{-1/2} = \frac{2}{\sqrt{x}}$ .

Combining these results, the derivative is:

$\frac{dy}{dx} = 6x + \frac{2}{\sqrt{x}}$

Step 2

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Answer

To find the second derivative, we differentiate the first derivative:

Starting with $\frac{dy}{dx} = 6x + \frac{2}{\sqrt{x}}$ , we differentiate:

For $6x$ , the derivative is $6$ .
For $\frac{2}{\sqrt{x}}$ , we rewrite it as $2x^{-1/2}$ . Therefore, its derivative is $-\frac{1}{2} \cdot 2x^{-3/2} = -\frac{1}{x^{3/2}}$ .

Combining these, we have:

$\frac{d^2y}{dx^2} = 6 - \frac{1}{x^{3/2}}$

Step 3

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Answer

To find the integral of $y$ , we integrate each term:

For $3x^2$ , the integral is $\frac{3}{3}x^3 = x^3$ .
For $4\sqrt{x}$ or $4x^{1/2}$ , we have: $4 \cdot \frac{1}{3/2}x^{3/2} = \frac{4 \cdot 2}{3}x^{3/2} = \frac{8}{3}x^{3/2}.$

Thus, the integral is:

$\int y \, dx = x^3 + \frac{8}{3}x^{3/2} + C,$ where $C$ is the constant of integration.