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Given the function f(x)=4(-4x^(2)-3x-5)^(3), find f^(')(x) in any form. Answer: f^(')(x)=

Given the function f(x)=4(−4x2−3x−5)3 f(x)=4\left(-4 x^{2}-3 x-5\right)^{3} , find f′(x) f^{\prime}(x) in any form.\newlineAnswer: f′(x)= f^{\prime}(x)=

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Q. Given the function f(x)=4(−4x2−3x−5)3 f(x)=4\left(-4 x^{2}-3 x-5\right)^{3} , find f′(x) f^{\prime}(x) in any form.\newlineAnswer: f′(x)= f^{\prime}(x)=
  1. Identify Functions: To find the derivative of the function f(x)=4(−4x2−3x−5)3f(x)=4(-4x^2-3x-5)^3, we will use the chain rule. The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.
  2. Find Outer Function Derivative: First, let's identify the outer function and the inner function. The outer function is u3u^3 where uu is the inner function, and the inner function is −4x2−3x−5-4x^2-3x-5.
  3. Find Inner Function Derivative: The derivative of the outer function u3u^3 with respect to uu is 3u23u^2. We will later substitute the inner function back into uu.
  4. Combine Inner Function Derivative: Now, we need to find the derivative of the inner function −4x2−3x−5-4x^2-3x-5 with respect to xx. The derivative of −4x2-4x^2 is −8x-8x, the derivative of −3x-3x is −3-3, and the derivative of a constant −5-5 is 00.
  5. Apply Chain Rule: Combining the derivatives of the terms in the inner function, we get the derivative of the inner function as −8x−3-8x - 3.
  6. Calculate Derivative: Now, we apply the chain rule. Multiply the derivative of the outer function, which is 3u23u^2, by the derivative of the inner function, which is −8x−3-8x - 3. Don't forget to substitute the inner function back into uu.
  7. Simplify Expression: The derivative of the function f(x)f(x) is then f′(x)=3×4×(−4x2−3x−5)2×(−8x−3)f'(x) = 3 \times 4 \times (-4x^2 - 3x - 5)^2 \times (-8x - 3). We multiply by 44 because it is a constant factor in the original function.
  8. Simplify Expression: The derivative of the function f(x)f(x) is then f′(x)=3×4×(−4x2−3x−5)2×(−8x−3)f'(x) = 3 \times 4 \times (-4x^2 - 3x - 5)^2 \times (-8x - 3). We multiply by 44 because it is a constant factor in the original function.Simplify the expression to get the final form of the derivative. f′(x)=12×(−4x2−3x−5)2×(−8x−3)f'(x) = 12 \times (-4x^2 - 3x - 5)^2 \times (-8x - 3).

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