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To find the first derivative of (2-3x)5 with respect to x.

Mathematics

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Answer

−15(2−3x)4-15(2-3x)^4

Step-by-step solution

5 steps
  1. Identify the function to differentiate.

    The problem asks to find the first derivative of (2−3x)5(2-3x)^5 with respect to xx.
    Let y=(2−3x)5y = (2-3x)^5.

  2. Apply the chain rule for differentiation.

    The chain rule states that if y=f(g(x))y = f(g(x)), then dydx=f′(g(x))⋅g′(x)\frac{dy}{dx} = f'(g(x)) \cdot g'(x).
    In this case, let g(x)=2−3xg(x) = 2-3x and f(u)=u5f(u) = u^5, where u=g(x)u = g(x).

  3. Differentiate f(u)f(u) with respect to uu.

    If f(u)=u5f(u) = u^5, then f′(u)=ddu(u5)=5u5−1=5u4f'(u) = \frac{d}{du}(u^5) = 5u^{5-1} = 5u^4.

  4. Differentiate g(x)g(x) with respect to xx.

    If g(x)=2−3xg(x) = 2-3x, then g′(x)=ddx(2−3x)=ddx(2)−ddx(3x)=0−3=−3g'(x) = \frac{d}{dx}(2-3x) = \frac{d}{dx}(2) - \frac{d}{dx}(3x) = 0 - 3 = -3.

  5. Multiply the results from Step 3 and Step 4, and substitute uu back.

    dydx=f′(g(x))⋅g′(x)\frac{dy}{dx} = f'(g(x)) \cdot g'(x) dydx=5(2−3x)4⋅(−3)\frac{dy}{dx} = 5(2-3x)^4 \cdot (-3) dydx=−15(2−3x)4\frac{dy}{dx} = -15(2-3x)^4

    The first derivative of (2−3x)5(2-3x)^5 is −15(2−3x)4\boxed{-15(2-3x)^4}.

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