NSC Mathematics Calculus: 7.1 Determine $f^{ ext{'}}(x)$ f

7.1 Determine $f^{ ext{'}}(x)$ from first principles if $f(x) = x^2 - 5$ - NSC Mathematics - Question 7 - 2017 - Paper 1

Question 7

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7.1 Determine $f^{ ext{'}}(x)$ from first principles if $f(x) = x^2 - 5$. 7.2 Determine the derivative of: $g(x) = 5x^3 - \frac{2x}{x^3}$. 7.3 Given: $h(x) = a x^2... show full transcript

Worked Solution & Example Answer:7.1 Determine $f^{ ext{'}}(x)$ from first principles if $f(x) = x^2 - 5$ - NSC Mathematics - Question 7 - 2017 - Paper 1

Step 1

Determine $f^{\text{'}}(x)$ from first principles if $f(x) = x^2 - 5$

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Answer

To find the derivative from first principles, we use the formula:

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

First, we compute $f(x + h)$ :

$f(x + h) = (x + h)^2 - 5 = (x^2 + 2xh + h^2 - 5)$

Now, substituting into the derivative formula:

$f^{\text{'}}(x) = \lim_{h \to 0} \frac{(x^2 + 2xh + h^2 - 5) - (x^2 - 5)}{h}$

Simplifying, we obtain:

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

Step 2

Determine the derivative of: $g(x) = 5x^3 - \frac{2x}{x^3}$

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Answer

We first simplify the function:

$g(x) = 5x^3 - 2x^{1-3} = 5x^3 - 2x^{-2}$

Now, we differentiate using the power rule:

$g^{\text{'}}(x) = 15x^2 + 4x^{-3} = 15x^2 + \frac{4}{x^3}$

Step 3

Given: $h(x) = ax^2, x > 0$. Determine the value of $a$ if it is given that $h^{\text{'}}(8) = h^{\text{'}}(4)$

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Answer

First, we compute the derivative:

$h^{\text{'}}(x) = 2ax$

Now, substituting values into the equation:

$h^{\text{'}}(8) = 2a(8) = 16a$ $h^{\text{'}}(4) = 2a(4) = 8a$

Setting them equal:

$16a = 8a$

Solving:

$16a - 8a = 0 \Rightarrow 8a = 0 \Rightarrow a = \frac{1}{8}$

7.1 Determine $f^{ ext{'}}(x)$ from first principles if $f(x) = x^2 - 5$ - NSC Mathematics - Question 7 - 2017 - Paper 1

Worked Solution & Example Answer:7.1 Determine $f^{ ext{'}}(x)$ from first principles if $f(x) = x^2 - 5$ - NSC Mathematics - Question 7 - 2017 - Paper 1

Determine $f^{\text{'}}(x)$ from first principles if $f(x) = x^2 - 5$

Determine the derivative of: $g(x) = 5x^3 - \frac{2x}{x^3}$

Given: $h(x) = ax^2, x > 0$. Determine the value of $a$ if it is given that $h^{\text{'}}(8) = h^{\text{'}}(4)$

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