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Apply the power rule (d)/(dx)(xⁿ) = nxⁿ−1.

Mathematics

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Answers

7 parts
a

dydx=1\frac{dy}{dx} = 1

b

dydx=−2x\frac{dy}{dx} = -2x

c

dydx=x2+x+1\frac{dy}{dx} = x^2 + x + 1

d

dydx=21x6−21x2+42x\frac{dy}{dx} = 21x^6 - 21x^2 + 42x

e

dydx=x3−x2+x−1\frac{dy}{dx} = x^3 - x^2 + x - 1

f

dydx=52x3/2\frac{dy}{dx} = \frac{5}{2} x^{3/2}

g

dydx=−25x−7/5\frac{dy}{dx} = -\frac{2}{5} x^{-7/5}

Step-by-step solution

9 steps

Here are the derivatives for each function:

Part a· 1 step

y=xy = x

  1. Apply the power rule ddx(xn)=nxn−1\frac{d}{dx}(x^n) = nx^{n-1}.

    dydx=1⋅x1−1\frac{dy}{dx} = 1 \cdot x^{1-1} dydx=1⋅x0\frac{dy}{dx} = 1 \cdot x^0 dydx=1⋅1\frac{dy}{dx} = 1 \cdot 1 dydx=1\boxed{\frac{dy}{dx} = 1}

Part b

dydx=−2x\frac{dy}{dx} = -2x

y=−x2+3y = -x^2 + 3

  1. Differentiate each term using the power rule and the constant rule.

    dydx=ddx(−x2)+ddx(3)\frac{dy}{dx} = \frac{d}{dx}(-x^2) + \frac{d}{dx}(3) dydx=−2x2−1+0\frac{dy}{dx} = -2x^{2-1} + 0 dydx=−2x\boxed{\frac{dy}{dx} = -2x}

Part c

dydx=x2+x+1\frac{dy}{dx} = x^2 + x + 1

y=x33+x22+xy = \frac{x^3}{3} + \frac{x^2}{2} + x

  1. Differentiate each term using the power rule.

    dydx=13⋅ddx(x3)+12⋅ddx(x2)+ddx(x)\frac{dy}{dx} = \frac{1}{3} \cdot \frac{d}{dx}(x^3) + \frac{1}{2} \cdot \frac{d}{dx}(x^2) + \frac{d}{dx}(x) dydx=13⋅3x3−1+12⋅2x2−1+1\frac{dy}{dx} = \frac{1}{3} \cdot 3x^{3-1} + \frac{1}{2} \cdot 2x^{2-1} + 1 dydx=x2+x+1\frac{dy}{dx} = x^2 + x + 1 dydx=x2+x+1\boxed{\frac{dy}{dx} = x^2 + x + 1}

Part d

dydx=21x6−21x2+42x\frac{dy}{dx} = 21x^6 - 21x^2 + 42x

y=3x7−7x3+21x2y = 3x^7 - 7x^3 + 21x^2

  1. Differentiate each term using the power rule.

    dydx=3⋅ddx(x7)−7⋅ddx(x3)+21⋅ddx(x2)\frac{dy}{dx} = 3 \cdot \frac{d}{dx}(x^7) - 7 \cdot \frac{d}{dx}(x^3) + 21 \cdot \frac{d}{dx}(x^2) dydx=3⋅7x7−1−7⋅3x3−1+21⋅2x2−1\frac{dy}{dx} = 3 \cdot 7x^{7-1} - 7 \cdot 3x^{3-1} + 21 \cdot 2x^{2-1} dydx=21x6−21x2+42x\frac{dy}{dx} = 21x^6 - 21x^2 + 42x dydx=21x6−21x2+42x\boxed{\frac{dy}{dx} = 21x^6 - 21x^2 + 42x}

Part e

dydx=x3−x2+x−1\frac{dy}{dx} = x^3 - x^2 + x - 1

y=x44−x33+x22−x+3y = \frac{x^4}{4} - \frac{x^3}{3} + \frac{x^2}{2} - x + 3

  1. Differentiate each term using the power rule and the constant rule.

    dydx=14⋅ddx(x4)−13⋅ddx(x3)+12⋅ddx(x2)−ddx(x)+ddx(3)\frac{dy}{dx} = \frac{1}{4} \cdot \frac{d}{dx}(x^4) - \frac{1}{3} \cdot \frac{d}{dx}(x^3) + \frac{1}{2} \cdot \frac{d}{dx}(x^2) - \frac{d}{dx}(x) + \frac{d}{dx}(3) dydx=14⋅4x4−1−13⋅3x3−1+12⋅2x2−1−1+0\frac{dy}{dx} = \frac{1}{4} \cdot 4x^{4-1} - \frac{1}{3} \cdot 3x^{3-1} + \frac{1}{2} \cdot 2x^{2-1} - 1 + 0 dydx=x3−x2+x−1\frac{dy}{dx} = x^3 - x^2 + x - 1 dydx=x3−x2+x−1\boxed{\frac{dy}{dx} = x^3 - x^2 + x - 1}

Part f

dydx=52x3/2\frac{dy}{dx} = \frac{5}{2} x^{3/2}

y=x5y = \sqrt{x^5}

  1. Rewrite the function using fractional exponents.

    y=x5/2y = x^{5/2}

  2. Apply the power rule.

    dydx=52x52−1\frac{dy}{dx} = \frac{5}{2} x^{\frac{5}{2} - 1} dydx=52x52−22\frac{dy}{dx} = \frac{5}{2} x^{\frac{5}{2} - \frac{2}{2}} dydx=52x3/2\frac{dy}{dx} = \frac{5}{2} x^{3/2} dydx=52x3/2\boxed{\frac{dy}{dx} = \frac{5}{2} x^{3/2}}

Part g

dydx=−25x−7/5\frac{dy}{dx} = -\frac{2}{5} x^{-7/5}

y=1x2/5y = \frac{1}{x^{2/5}}

  1. Rewrite the function using negative exponents.

    y=x−2/5y = x^{-2/5}

  2. Apply the power rule.

    dydx=−25x−25−1\frac{dy}{dx} = -\frac{2}{5} x^{-\frac{2}{5} - 1} dydx=−25x−25−55\frac{dy}{dx} = -\frac{2}{5} x^{-\frac{2}{5} - \frac{5}{5}} dydx=−25x−7/5\frac{dy}{dx} = -\frac{2}{5} x^{-7/5} dydx=−25x−7/5\boxed{\frac{dy}{dx} = -\frac{2}{5} x^{-7/5}}
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