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Differentiate the following: a) y = 3x⁻³ b) y = (4)/(x2)

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2 parts
a

−9x−4-9x^{-4}

b

−8x−3-8x^{-3}

Step-by-step solution

5 steps

2) Differentiate the following

Part a· 2 steps

y=3x−3y = 3x^{-3}

  1. Apply the power rule for differentiation, which states that if y=axny = ax^n, then dydx=naxn−1\frac{dy}{dx} = nax^{n-1}.

    Here, a=3a=3 and n=−3n=-3.

    dydx=(−3)⋅3x−3−1\frac{dy}{dx} = (-3) \cdot 3x^{-3-1}

  2. Simplify the expression.

    dydx=−9x−4\frac{dy}{dx} = -9x^{-4}

    The derivative is −9x−4\boxed{-9x^{-4}}.

Part b

−8x−3-8x^{-3}

y=4x2y = \frac{4}{x^2}

  1. Rewrite the function in the form axnax^n.

    y=4x−2y = 4x^{-2}

  2. Apply the power rule for differentiation.

    Here, a=4a=4 and n=−2n=-2.

    dydx=(−2)⋅4x−2−1\frac{dy}{dx} = (-2) \cdot 4x^{-2-1}

  3. Simplify the expression.

    dydx=−8x−3\frac{dy}{dx} = -8x^{-3}

    The derivative is −8x−3\boxed{-8x^{-3}}.

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