Calculus: Derivatives I

Trigonometric Functions

The derivatives for trigonometric functions can be derived from their complex exponential forms, as well as through the cunning yet tedious use of trigonometric identities. However, for now it is simple enough to show the derivatives for the three basic trigonometric functions.

$$\dfrac{d}{dx} \sin(x) = \cos(x)$$

$$\dfrac{d}{dx} \cos(x) = -\sin(t)$$

$$\dfrac{d}{dx} \tan(x) = \sec^2{x}$$

Problems

  1. Differentiate with respect to $x$: $y= \sin(x) + \cos(x) - \tan(x)$

    $\dfrac{d}{dx}y = \dfrac{d}{dx}\left(\sin(x) + \cos(x) - \tan(x)\right) \\ \dfrac{dy}{dx} = \dfrac{d}{dx}\sin(x) + \dfrac{d}{dx}\cos(x) - \dfrac{d}{dx}\tan(x) \\ \dfrac{dy}{dx} = \cos(x) + -\sin(x) - \sec^2(x) \\ $
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  2. Differentiate with respect to $x$: $y= \csc(x) - \dfrac{1}{\sin(x)} + \cot(x)$

    $\dfrac{d}{dx}y = \dfrac{d}{dx}\left(\csc(x) - \dfrac{1}{\sin(x)} + \cot(x)\right) \\ \dfrac{dy}{dx} = \dfrac{d}{dx}\csc(x) - \dfrac{d}{dx}\dfrac{1}{\sin(x)} + \dfrac{d}{dx}\cot(x) \\ \dfrac{dy}{dx} = \dfrac{d}{dx}\csc(x) - \dfrac{d}{dx}\csc(x) + \dfrac{d}{dx}\cot(x) \\ \dfrac{dy}{dx} = \dfrac{d}{dx}\cot(x) \\ \dfrac{dy}{dx} = -\csc^2(x) \\ $
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  3. Differentiate with respect to $x$: $y= \sec(x) + \tan(x) + \dfrac{1}{\sin(x)}$

    $\dfrac{d}{dx}y = \dfrac{d}{dx}\left(\sec(x) + \tan(x) + \dfrac{1}{\sin(x)}\right) \\ \dfrac{dy}{dx} = \dfrac{d}{dx}\sec(x) + \dfrac{d}{dx}\tan(x) + \dfrac{d}{dx}\dfrac{1}{\sin(x)} \\ \dfrac{dy}{dx} = \dfrac{d}{dx}\sec(x) + \dfrac{d}{dx}\tan(x) + \dfrac{d}{dx}\csc(x) \\ \dfrac{dy}{dx} = \tan(x)\sec(x) + \sec^2(x) -\cot(x)\csc(x) \\ $
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  4. Differentiate with respect to $x$: $y= \sin(x)\left(\sin(x) + \cos^2(x)\csc(x)\right)$

    $\dfrac{d}{dx}y = \dfrac{d}{dx}\left(\sin(x)\left(\sin(x) + \cos^2(x)\csc(x)\right)\right) \\ \dfrac{dy}{dx} = \dfrac{d}{dx}\left(\sin^2(x) + \sin(x)\cos^2(x)\csc(x)\right) \\ \dfrac{dy}{dx} = \dfrac{d}{dx}\left(\sin^2(x) + \cos^2(x)\right) \\ \dfrac{dy}{dx} = \dfrac{d}{dx}1 \\ \dfrac{dy}{dx} = 0 \\ $
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  5. Differentiate with respect to $x$: $y= \sum\limits_{i=1}^{100}\sin(x)$

    $\dfrac{d}{dx}y = \dfrac{d}{dx}\sum\limits_{i=1}^{100}\sin(x) \\ \dfrac{dy}{dx} = \sum\limits_{i=1}^{100}\dfrac{d}{dx}\sin(x) \\ \dfrac{dy}{dx} = \sum\limits_{i=1}^{100}\cos(x) \\ \dfrac{dy}{dx} = 100\cos(x) $
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  6. Differentiate with respect to $x$: $y = \sec^2(\alpha)$

    $\dfrac{d}{dx}y = \dfrac{d}{dx}\sec^2(\alpha) \\ \dfrac{dy}{dx} = 0 \\ $
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  7. Differentiate with respect to $\alpha$: $y= \sec^2(\rho)$

    $\dfrac{d}{d\alpha}y = \dfrac{d}{d\alpha}\sec^2(\rho) \\ \dfrac{dy}{d\alpha} = 0 \\ $
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  8. Simplify: $y = -\sin(x)\dfrac{d}{dx}\cos(x) -\cos(x)\dfrac{d}{dx}\sin(x)$

    $y = -\sin(x)\dfrac{d}{dx}\cos(x) -\cos(x)\dfrac{d}{dx}\sin(x) \\ y = -\sin(x)\left(-\sin(x)\right) - \cos^{2}(x) \\ y = \sin^2(x) - \cos^{2}(x) \\ $
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  9. Simplify: $y = -\dfrac{d}{dx}\left(\dfrac{d}{dx} \sin(x)\right)$

    $y = -\dfrac{d}{dx}\left(\dfrac{d}{dx} \sin(x)\right) \\ y = -\dfrac{d}{dx}\cos(x) \\ y = -\left(-\sin(x)\right) \\ y = \sin(x) \\ $
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  10. Differentiate with respect to $x$: $y = \sin^2\left(\dfrac{x}{2}\right) - \cos^2\left(\dfrac{x}{2}\right)$

    $\dfrac{d}{dx}y = \dfrac{d}{dx}\left( \sin^2\left(\dfrac{x}{2}\right) - \cos^2\left(\dfrac{x}{2}\right) \right)\\ \dfrac{dy}{dx} = \dfrac{d}{dx}\left( -\cos(x) \right) \\ \dfrac{dy}{dx} = -\dfrac{d}{dx}\cos(x) \\ \dfrac{dy}{dx} = -(-\sin(x)) \\ \dfrac{dy}{dx} = \sin(x) \\ $
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