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T1202s Maths N3 2.1. Differentiate the following function from the first principle: \[ f(x)=-2 t \pi-9 t^{2} x^{2}+4 t^{3} x \] 2.2. Determine \( \frac{d y}{d x} \) of the following function by making use of the rules of differentiation. Att flnat answers must be whth a poshtive exponent and in a surd form whore appticatole. \[ y=\frac{2}{3 x^{4}}-\frac{1}{2 x^{3}}-4 \sqrt[3]{x}+8 \] 2.3. Make use of differentiation to determine the co-ordinates of the turming points of the given function below. \[ f(x)=x\left(x^{2}-6 x+9\right) \] Question 3 3.1. In the following trigonometric equation without the use of a calculator solve for \( x \); where \( 0^{\circ} \leq x \leq 360^{\circ} \) \[ \sqrt{2} \sin x=1 \] 3.2. Prove that: \( (\tan \alpha+\cot \alpha)^{2}=\operatorname{cosec}^{2} \alpha \cdot \sec ^{2} \alpha \) (4)

Ask by Hilton Hodgson. in South Africa
Mar 12,2025

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**2.1 Differentiation from First Principle** The derivative of the function \( f(x) = -2t\pi - 9t^2x^2 + 4t^3x \) is: \[ f'(x) = -18t^2x + 4t^3 \] **2.2 Differentiation Using Rules** The derivative of \( y = \frac{2}{3x^4} - \frac{1}{2x^3} - 4\sqrt[3]{x} + 8 \) is: \[ \frac{dy}{dx} = -\frac{8}{3}x^{-5} + \frac{3}{2}x^{-4} - \frac{4}{3}x^{-2/3} \] **2.3 Turning Points** The turning points of \( f(x) = x(x^2 - 6x + 9) \) are at \( (0, 0) \) and \( (3, 0) \). **3.1 Trigonometric Equation Solution** The solutions for \( \sqrt{2} \sin x = 1 \) in the interval \( 0^\circ \leq x \leq 360^\circ \) are: \[ x = 45^\circ \quad \text{and} \quad x = 135^\circ \] **3.2 Trigonometric Identity Proof** \[ (\tan \alpha + \cot \alpha)^2 = \csc^2 \alpha \cdot \sec^2 \alpha \] Proof: \[ (\tan \alpha + \cot \alpha)^2 = \left(\frac{\sin \alpha}{\cos \alpha} + \frac{\cos \alpha}{\sin \alpha}\right)^2 = \frac{1}{\sin^2 \alpha \cos^2 \alpha} = \csc^2 \alpha \cdot \sec^2 \alpha \]

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Bonus Knowledge

Differentiation, especially through first principles, dates back to the pioneering work of mathematicians like Isaac Newton and Gottfried Wilhelm Leibniz in the 17th century. They developed calculus around the same time but independently, fundamentally transforming mathematics and our understanding of change. The first principles method—finding a derivative using limits—encapsulates this conceptual innovation, allowing us to explore how functions behave at infinitesimally small intervals. When tackling differentiation, remember to check for common mistakes! A classic error is neglecting the chain rule when dealing with composite functions. Also, keep an eye on your signs, especially when dealing with negative exponents or factoring. Always double-check your algebraic manipulation, as it's easy to miss an opportunity for simplification, especially when roots or surds are involved. These little checks can save you from headaches down the line!

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