6.1 Determine $f'(x)$ using FIRST PRINCIPLES if $f(x) = \frac{7}{2}x + 5$. 6.2 Determine: 6.2.1 $f'(x)$ if $f(x) = -8\pi$ 6.2.2 $ \frac{dy}{dx} $ if $y = \frac{x^4 + 9x}{x^2}$ 6.2.3 $D_1\left[ \left( \sqrt{x + 8} \right) \right]$ 6.3 Given: $g(x) = 3x^2 + 9x$ 6.3.1 Determine $g'(x)$. 6.3.2 Determine the gradient of the tangent to the curve of $g$ at the point where $x = -3$. 6.3.3 Hence, determine the equation of the tangent to the curve of $g$ where $x = -3$.

NSC Technical Mathematics Differential Calculus and Integration: 6.1 Determine $f'(x)$ using FIRS

6.1 Determine $f'(x)$ using FIRST PRINCIPLES if $f(x) = \frac{7}{2}x + 5$ - NSC Technical Mathematics - Question 6 - 2023 - Paper 1

Question 6

$6.1-Determine-$f'(x)$-using-FIRST-PRINCIPLES-if-$f(x)-=-\frac{7}{2}x-+-5$-NSC Technical Mathematics-Question 6-2023-Paper 1.png$

6.1 Determine $f'(x)$ using FIRST PRINCIPLES if $f(x) = \frac{7}{2}x + 5$. 6.2 Determine: 6.2.1 $f'(x)$ if $f(x) = -8\pi$ 6.2.2 $ \frac{dy}{dx} $ if $y = \frac... show full transcript

Worked Solution & Example Answer:6.1 Determine $f'(x)$ using FIRST PRINCIPLES if $f(x) = \frac{7}{2}x + 5$ - NSC Technical Mathematics - Question 6 - 2023 - Paper 1

Step 1

Determine $f'(x)$ using FIRST PRINCIPLES if $f(x) = \frac{7}{2}x + 5$

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Answer

To find the derivative using first principles, we apply the formula: $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$ Substituting our function: $f'(x) = \lim_{h \to 0} \frac{\left(\frac{7}{2}(x+h) + 5\right) - \left(\frac{7}{2}x + 5\right)}{h}$ Simplifying: $f'(x) = \lim_{h \to 0} \frac{\frac{7}{2}h}{h} = \frac{7}{2}$

Step 2

Determine $f'(x)$ if $f(x) = -8\pi$

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Answer

Since $f(x) = -8\pi$ is a constant function, its derivative is: $f'(x) = 0$

Step 3

Determine $ \frac{dy}{dx} $ if $y = \frac{x^4 + 9x}{x^2}$

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Answer

We simplify first: $y = x^2 + \frac{9}{x}$ Now, using the quotient rule: $\frac{dy}{dx} = \frac{(2x)(x^2) - (x^4 + 9x)(2x)}{(x^2)^2}$ Expanding gives: $\frac{dy}{dx} = \frac{2x^3 - (2x^5 + 18x^2)}{x^4} = \frac{-2x^5 - 16x^2}{x^4} = -2x - \frac{16}{x^2}$

Step 4

Determine $D_1\left[ \left( \sqrt{x + 8} \right) \right]$

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Answer

To find the first derivative: $D_1\left(\sqrt{x + 8}\right) = \frac{1}{2\sqrt{x + 8}}$

Step 5

Determine $g'(x)$ if $g(x) = 3x^2 + 9x$

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Answer

Using basic differentiation rules, we find: $g'(x) = 6x + 9$

Step 6

Determine the gradient of the tangent to the curve of $g$ at the point where $x = -3$

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Answer

Substituting $x = -3$ into $g'(x)$ gives: $g'(-3) = 6(-3) + 9 = -18 + 9 = -9$

Step 7

Hence, determine the equation of the tangent to the curve of $g$ where $x = -3$

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Answer

The point on the curve is: $g(-3) = 3(-3)^2 + 9(-3) = 27 - 27 = 0$ The point is $(-3, 0)$ . Using point-slope form: $y - 0 = -9(x + 3)$ Thus, the equation of the tangent is: $y = -9x - 27$

6.1 Determine $f'(x)$ using FIRST PRINCIPLES if $f(x) = \frac{7}{2}x + 5$ - NSC Technical Mathematics - Question 6 - 2023 - Paper 1

Worked Solution & Example Answer:6.1 Determine $f'(x)$ using FIRST PRINCIPLES if $f(x) = \frac{7}{2}x + 5$ - NSC Technical Mathematics - Question 6 - 2023 - Paper 1

Determine $f'(x)$ using FIRST PRINCIPLES if $f(x) = \frac{7}{2}x + 5$

Determine $f'(x)$ if $f(x) = -8\pi$

Determine \( \frac{dy}{dx} \) if $y = \frac{x^4 + 9x}{x^2}$

Determine $D_1\left[ \left( \sqrt{x + 8} \right) \right]$

Determine $g'(x)$ if $g(x) = 3x^2 + 9x$

Determine the gradient of the tangent to the curve of $g$ at the point where $x = -3$

Hence, determine the equation of the tangent to the curve of $g$ where $x = -3$

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