CALCULUS
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Master calculus, the branch of mathematics focused on change and motion. Learn key concepts like limits, derivatives, integrals, and differential equations. Explore real-world applications in physics, engineering, economics, and beyond. Perfect for students and professionals aiming to deepen their understanding of continuous functions and mathematical modeling
Derivative of sinθ using the First Principle
Derivative of \( sinθ \) using the First Principle Let \(y\) = \( sinθ \) ∴ \(y + δy\) = \( sin(θ + δθ) \) ∴ \(δy\) = \( sin(θ + δθ) \) - \( sinθ \)From Trigonometry , we have \( sin(A-B) \) = 2.\( sin \dfrac {(A-B)}{2} \).\( cos \dfrac {(A+B)}{2} \)Using the above rule, we get\(δy\) = 2.\( sin \dfrac {(θ+δθ - θ)}{2} \).\( cos \dfrac {(θ+δθ + θ)}{2} \) or \(δy\) = 2\( cos (θ+ \dfrac{δθ}{2}) \) . \( sin( \dfrac {δθ}{2}) \)∴ \(\dfrac {δy}{δθ}\) = 2 \( \dfrac { cos (θ+ \dfrac{δθ}{2}) sin( \dfrac {δθ}{2}) } {δθ}\)or \(\dfrac {δy}{δθ}\) = 2 \(cos (θ+ \dfrac{δθ}{2}) \) \( \dfrac { sin( \dfrac {δθ}{2}) } {δθ}\) ∴ \(\dfrac […] READ MORE...
Derivative of cosθ using the First Principle
Derivative of \( cosθ \) using the First Principle Let \(y\) = \( cosθ \) ∴ \(y + δy\) = \( cos(θ + δθ) \) ∴ \(δy\) = \( cos(θ + δθ) \) - \( cosθ \)From Trigonometry , we have \( cos(A-B) \) = -2.\( sin \dfrac {(A+B)}{2} \).\( sin \dfrac {(A-B)}{2} \)Using the above rule, we get\(δy\) = -2.\( sin \dfrac {(θ+δθ + θ)}{2} \).\( sin \dfrac {(θ+δθ - θ)}{2} \) or \(δy\) = -2\( sin (θ+ \dfrac{δθ}{2}) \) . \( sin( \dfrac {δθ}{2}) \)∴ \(\dfrac {δy}{δθ}\) = -2 \( \dfrac { sin (θ+ \dfrac{δθ}{2}) sin( \dfrac {δθ}{2}) } {δθ}\)or \(\dfrac {δy}{δθ}\) = -2 \(sin (θ+ \dfrac{δθ}{2}) \) \( \dfrac { sin( \dfrac {δθ}{2}) } {δθ}\) ∴ \(\dfrac […] READ MORE...