NSC Mathematics Calculus: Bepaal $f'(x)$ vanuit eerste beg

Bepaal $f'(x)$ vanuit eerste beginsels indien dit gegeven wordt dat $f(x) = 3x^2$ - NSC Mathematics - Question 8 - 2021 - Paper 1

Question 8

Bepaal $f'(x)$ vanuit eerste beginsels indien dit gegeven wordt dat $f(x) = 3x^2$. 8.2 Bepaal: 8.2.1 $f'(x)$ indien $f(x) = x^2 - 3 + \frac{9}{x^2}$. 8.2.2 $g'(x)... show full transcript

Worked Solution & Example Answer:Bepaal $f'(x)$ vanuit eerste beginsels indien dit gegeven wordt dat $f(x) = 3x^2$ - NSC Mathematics - Question 8 - 2021 - Paper 1

Step 1

Bepaal $f'(x)$ vanuit eerste beginsels indien dit gegeven wordt dat $f(x) = 3x^2$

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Answer

To find the derivative using the first principles, we use the definition of the derivative:

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

Substitute for $f(x)$ : $f'(x) = \lim_{h \to 0} \frac{3(x+h)^2 - 3x^2}{h}$
Expand $(x+h)^2$ : $= \lim_{h \to 0} \frac{3(x^2 + 2xh + h^2) - 3x^2}{h}$ $= \lim_{h \to 0} \frac{3x^2 + 6xh + 3h^2 - 3x^2}{h}$
Simplify: $= \lim_{h \to 0} \frac{6xh + 3h^2}{h}$ $= \lim_{h \to 0} (6x + 3h)$
Evaluate the limit as $h$ approaches 0: $f'(x) = 6x.$

Step 2

Bepaal $f'(x)$ indien $f(x) = x^2 - 3 + \frac{9}{x^2}$

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Answer

Using standard rules of differentiation:

Differentiate each term:
- The derivative of $x^2$ is $2x$ .
- The derivative of $-3$ is $0$ .
- For $rac{9}{x^2}$ , rewrite as $9x^{-2}$ , then differentiate to get $-18x^{-3}$ .

Putting it all together:

$f'(x) = 2x - 18x^{-3}.$

Step 3

Bepaal $g'(x)$ indien $g(x) = (\sqrt{x+3})(\sqrt{x-1})$

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Answer

To find the derivative of $g(x)$ , we will apply the product rule:

Define the functions: $u = \sqrt{x+3}$ and $v = \sqrt{x-1}$ .
Use the product rule: $g'(x) = u'v + uv'.$
Differentiating $u$ and $v$ :
- $u' = \frac{1}{2\sqrt{x+3}}$ and $v' = \frac{1}{2\sqrt{x-1}}$ .
Substitute: $g'(x) = \frac{1}{2\sqrt{x+3}}\sqrt{x-1} + \sqrt{x+3}\frac{1}{2\sqrt{x-1}}.$

The answer can be further simplified if needed.

Bepaal $f'(x)$ vanuit eerste beginsels indien dit gegeven wordt dat $f(x) = 3x^2$ - NSC Mathematics - Question 8 - 2021 - Paper 1

Worked Solution & Example Answer:Bepaal $f'(x)$ vanuit eerste beginsels indien dit gegeven wordt dat $f(x) = 3x^2$ - NSC Mathematics - Question 8 - 2021 - Paper 1

Bepaal $f'(x)$ vanuit eerste beginsels indien dit gegeven wordt dat $f(x) = 3x^2$

Bepaal $f'(x)$ indien $f(x) = x^2 - 3 + \frac{9}{x^2}$

Bepaal $g'(x)$ indien $g(x) = (\sqrt{x+3})(\sqrt{x-1})$

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