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For the following equation, evaluate f^(')(1). f(x)=-2x^(3)+5 Answer:

For the following equation, evaluate f′(1) f^{\prime}(1) .\newlinef(x)=−2x3+5 f(x)=-2 x^{3}+5 \newlineAnswer:

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Composition of linear and quadratic functions: find a valueright chevron icon

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Q. For the following equation, evaluate f′(1) f^{\prime}(1) .\newlinef(x)=−2x3+5 f(x)=-2 x^{3}+5 \newlineAnswer:
  1. Identify Function and Derivative: Identify the function and the derivative to be evaluated.\newlineWe are given the function f(x)=−2x3+5f(x) = -2x^3 + 5 and we need to find its derivative f′(x)f'(x) at x=1x = 1.
  2. Calculate Derivative of Function: Calculate the derivative of the function f(x)f(x). The derivative of f(x)=−2x3+5f(x) = -2x^3 + 5 with respect to xx is f′(x)=ddx(−2x3)+ddx(5)f'(x) = \frac{d}{dx}(-2x^3) + \frac{d}{dx}(5). Using the power rule for differentiation, the derivative of −2x3-2x^3 is −6x2-6x^2. The derivative of a constant, 55, is 00. So, f′(x)=−6x2+0f'(x) = -6x^2 + 0, which simplifies to f′(x)=−6x2f'(x) = -6x^2.
  3. Evaluate Derivative at x = 11: Evaluate the derivative at x=1x = 1.\newlineSubstitute x=1x = 1 into the derivative f′(x)=−6x2f'(x) = -6x^2.\newlinef′(1)=−6(1)2f'(1) = -6(1)^2\newlinef′(1)=−6(1)f'(1) = -6(1)\newlinef′(1)=−6f'(1) = -6

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