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7.1 Determine $f'(x)$ from first principles if $f(x)=2x^2-1$ - NSC Mathematics - Question 7 - 2020 - Paper 1

Question 7

7.1 Determine $f'(x)$ from first principles if $f(x)=2x^2-1$. 7.2 Determine: 7.2.1 $\frac{d}{dx}(\sqrt{x^2+x^3})$ 7.2.2 $f'(x)$ if $f(x)=\frac{4x^2-9}{4x+6}; \... show full transcript

Worked Solution & Example Answer:7.1 Determine $f'(x)$ from first principles if $f(x)=2x^2-1$ - NSC Mathematics - Question 7 - 2020 - Paper 1

Step 1

Determine $f'(x)$ from first principles if $f(x)=2x^2-1$

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Answer

To find $f'(x)$ from first principles, we use the definition of the derivative:

$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

Starting with $f(x) = 2x^2 - 1$ , we first calculate $f(x + h)$ :

$f(x+h) = 2(x+h)^2 - 1 = 2(x^2 + 2xh + h^2) - 1 = 2x^2 + 4xh + 2h^2 - 1$

Now we substitute into the limit:

$f'(x) = \lim_{h \to 0} \frac{(2x^2 + 4xh + 2h^2 - 1) - (2x^2 - 1)}{h}$

This simplifies to:

$f'(x) = \lim_{h \to 0} \frac{4xh + 2h^2}{h} = \lim_{h \to 0} (4x + 2h)$

Evaluating the limit as $h$ approaches 0 gives:

$f'(x) = 4x$

Step 2

Determine: $\frac{d}{dx}(\sqrt{x^2 + x^3})$

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Answer

To differentiate $y = \sqrt{x^2 + x^3}$ , we apply the chain rule:

$\frac{dy}{dx} = \frac{1}{2\sqrt{x^2 + x^3}} \cdot \frac{d}{dx}(x^2 + x^3)$

Calculating the derivative of the inside:

$\frac{d}{dx}(x^2 + x^3) = 2x + 3x^2$

Therefore:

$\frac{dy}{dx} = \frac{1}{2\sqrt{x^2 + x^3}} (2x + 3x^2) = \frac{2x + 3x^2}{2\sqrt{x^2 + x^3}}$

Step 3

Determine: $f'(x)$ if $f(x)=\frac{4x^2-9}{4x+6}; \quad x=\frac{3}{2}$

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Answer

To differentiate $f(x) = \frac{4x^2 - 9}{4x + 6}$ , we use the quotient rule:

$f'(x) = \frac{(g(x)f'(x) - f(x)g'(x))}{(g(x))^2}$

where $f(x) = 4x^2 - 9$ and $g(x) = 4x + 6$ . Therefore:

$f'(x) = 8x$
$g'(x) = 4$

Now substituting into the quotient rule:

$f'(x) = \frac{(4x + 6)(8x) - (4x^2 - 9)(4)}{(4x + 6)^2}$

We evaluate this at $x = \frac{3}{2}$ :

$f'\left(\frac{3}{2}\right) = 1$

7.1 Determine $f'(x)$ from first principles if $f(x)=2x^2-1$ - NSC Mathematics - Question 7 - 2020 - Paper 1

Worked Solution & Example Answer:7.1 Determine $f'(x)$ from first principles if $f(x)=2x^2-1$ - NSC Mathematics - Question 7 - 2020 - Paper 1

Determine $f'(x)$ from first principles if $f(x)=2x^2-1$

Determine: $\frac{d}{dx}(\sqrt{x^2 + x^3})$

Determine: $f'(x)$ if $f(x)=\frac{4x^2-9}{4x+6}; \quad x=\frac{3}{2}$

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