Calculus: Derivatives I

Polynomials

Taking the derivative of a polynomial is the simplest of simple things to do. The derivative of a polynomial is as follows:

$$\dfrac{d}{dx} x^n = nx^{n-1}$$

Remember that $\dfrac{1}{x^n} = x^{-n}$. For the derivation of this rule, see the Definition section of Derivatives I.

Problems

  1. Take the derivative with respect to $x$: $f(x) = 4x + 2$

    $f(x) = 4x + 2 \\ \dfrac{d}{dx} f(x) = \dfrac{d}{dx}(4x + 2) \\ f'(x) = 4 + 0 \\ f'(x) = 4 \\$

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  2. Differentiate with respect to x: $y = x^2$
    $\dfrac{d}{dx}y = \dfrac{d}{dx} x^2 \\ \dfrac{dy}{dx} = 2 \cdot x^{2-1} \\ \dfrac{dy}{dx} = 2x \\$
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  3. Take the derivative with respect to $x$: $y = 4x^2 + 3x + 5$

    $y = 4x^2 + 3x + 5 \\ \dfrac{d}{dx}y = \dfrac{d}{dx}(4x^2 + 3x + 5) \\ \dfrac{dy}{dx} = 8x + 3 + 0 \\ \dfrac{dy}{dx} = 8x + 3 \\ $

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  4. Differentiate with respect to $x$: $y = ax^2 + bx + c$

    $y = ax^2 + bx + c \\ \dfrac{d}{dx}y = \dfrac{d}{dx}(ax^2 + bx + c) \\ \dfrac{dy}{dx} = 2ax + b \\$

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  5. Differentiate with respect to $x$: $y = b^2 - 4ac$

    $y = b^2 - 4ac \\ \dfrac{d}{dx}y = \dfrac{d}{dx}(b^2 - 4ac) \\ \dfrac{dy}{dx} = 0 \\$

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  6. Differentiate with respect to $x$: $y = \dfrac{9}{x}$

    $y = \dfrac{9}{x} \\ \dfrac{d}{dx}y = \dfrac{d}{dx}\dfrac{9}{x} \\ \dfrac{dy}{dx} = \dfrac{d}{dx}9x^{-1} \\ \dfrac{dy}{dx} = -9x^{-2} \\ \dfrac{dy}{dx} = -\dfrac{9}{x^2} \\$

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  7. Differentiate with respect to $x$: $y=4x^4 + 3x^3 - 2$

    $\dfrac{d}{dx}y = \dfrac{d}{dx} \left( 4x^4 + 3x^3 - 2 \right) \\ \dfrac{dy}{dx} = \dfrac{d}{dx} 4x^4 + \dfrac{d}{dx}3x^3 - \dfrac{d}{dx}2 \\ \dfrac{dy}{dx} = 4\cdot4x^{4-1} + 3\cdot3x^{3-1} - 0 \\ \dfrac{dy}{dx} = 16x^{3} + 9x^{2} \\$
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  8. Differentiate with respect to $x$: $y=ax^b, b \neq 0$
    $\dfrac{d}{dx} y = \dfrac{d}{dx} ax^b\\ \dfrac{dy}{dx} = b\cdot ax^{b-1}\\ \dfrac{dy}{dx} = bax^{b-1}\\$
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  9. Differentiate with respect to $x$: $y=x^e$

    $\dfrac{d}{dx}y = \dfrac{d}{dx}x^e \\ \dfrac{dy}{dx} = ex^{e-1} \\$
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  10. Differentiate with respect to $x$: $y=2x^{-2} -2x^{-1} + 0 + 3x - 2x^{2}$

    $\dfrac{d}{dx}y = \dfrac{d}{dx}\left(2x^{-2} -2x^{-1} + 0 + 3x - 2x^{2}\right) \\ \dfrac{dy}{dx} = \dfrac{d}{dx}2x^{-2} -\dfrac{d}{dx}2x^{-1} + \dfrac{d}{dx}0 + \dfrac{d}{dx}3x - \dfrac{d}{dx}2x^{2} \\ \dfrac{dy}{dx} = -2\cdot2x^{-2-1} -(-1)\cdot2x^{-1-1} + 0 + 1\cdot3x^{0-1} - 2\cdot2x^{2-1} \\ \dfrac{dy}{dx} = -4x^{-3} + 2x^{-2} + 3 - 4x \\ $
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  11. Differentiate with respect to $x$: $y=\dfrac{2}{x}$

    $\dfrac{d}{dx}y = \dfrac{d}{dx}\dfrac{2}{x} \\
    \dfrac{dy}{dx} = \dfrac{d}{dx}2x^{-1} \\
    \dfrac{dy}{dx} = -1\cdot2x^{-1-1} \\
    \dfrac{dy}{dx} = -2x^{-2} \\
    \dfrac{dy}{dX} = \dfrac{-2}{x^2}\\
    $

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  12. Differentiate with respect to $x$: $y=\dfrac{2+x}{x^2}$

    $\dfrac{d}{dx}y = \dfrac{d}{dx}\dfrac{2+x}{x^2} \\ \dfrac{dy}{dx} = \dfrac{d}{dx}\left(\dfrac{2}{x^2} + \dfrac{x}{x^2}\right) \\ \dfrac{dy}{dx} = \dfrac{d}{dx}\left(2x^{-2} + x^{-1}\right) \\ \dfrac{dy}{dx} = \dfrac{d}{dx}2x^{-2} + \dfrac{d}{dx}x^{-1} \\ \dfrac{dy}{dx} = -2\cdot2x^{-2-1} + (-1)\cdot x^{-1-1} \\ \dfrac{dy}{dx} = -4x^{-3} - x^{-2} \\ \dfrac{dy}{dx} = \dfrac{-4 - x}{x^3} \\ $
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  13. Differentiate with respect to $\rho$: $y= \rho + \gamma^2 \rho + x - x\phi\rho^2$

    $\dfrac{d}{d\rho}y = \dfrac{d}{d\rho}\left( \rho + \gamma^2 \rho + x - x\phi\rho^2 \right) \\ \dfrac{dy}{d\rho} = \dfrac{d}{d\rho}\rho + \dfrac{d}{d\rho}\gamma^2 \rho + \dfrac{d}{d\rho}x - \dfrac{d}{d\rho}x\phi\rho^2 \\ \dfrac{dy}{d\rho} = 1\cdot \rho^{1-1} + 1\cdot\gamma^2 \rho^{1-1} + 0 - 2\cdot x\phi\rho^{2-1} \\ \dfrac{dy}{d\rho} = 1 + \gamma^2 - 2x\phi\rho \\ $
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