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3-5 Derivatives of Trigonometric Functions

Learning Targets

You will be able to

Concepts / Definitions

\(\frac{d}{dx}\sin(x) = \cos(x)\) \(\frac{d}{dx}\cos(x) = -\sin(x)\) \(\frac{d}{dx}\tan(x) = \sec^2(x)\) \(\frac{d}{dx}\cot(x) = -\csc^2(x)\) \(\frac{d}{dx}\sec(x) = \sec(x)\tan(x)\) \(\frac{d}{dx}\csc(x) = -\csc(x)\cot(x)\)

Proof of Derivative of Sine

\(f'(x) = \lim_{h\to 0} \frac{\sin(x+h) - \sin x}{h}\) \(f'(x) = \lim_{h\to 0} \frac{\sin x \cos h + \cos x \sin h - \sin x}{h}\) \(f'(x) = \lim_{h\to 0} \frac{\sin x (\cos h -1)}{h} + \lim_{h\to 0}\frac{\cos x \sin h}{h}\) \(f'(x) = \cos x\)

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