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Given the function f(x)=-3(4x^(2)-7x)^(5), find f^(')(x) in any form. Answer: f^(')(x)=

Given the function f(x)=−3(4x2−7x)5 f(x)=-3\left(4 x^{2}-7 x\right)^{5} , find f′(x) f^{\prime}(x) in any form.\newlineAnswer: f′(x)= f^{\prime}(x)=

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Full solution

Q. Given the function f(x)=−3(4x2−7x)5 f(x)=-3\left(4 x^{2}-7 x\right)^{5} , 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)=−3(4x2−7x)5f(x)=-3(4x^{2}-7x)^{5}, 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 u5u^5, where uu is the inner function. The inner function is 4x2−7x4x^2 - 7x.
  3. Find Inner Function Derivative: Now, we will find the derivative of the outer function with respect to the inner function uu. The derivative of u5u^5 with respect to uu is 5u45u^4.
  4. Apply Chain Rule: Next, we will find the derivative of the inner function 4x2−7x4x^2 - 7x with respect to xx. The derivative of 4x24x^2 is 8x8x, and the derivative of −7x-7x is −7-7. So, the derivative of the inner function is 8x−78x - 7.
  5. Simplify Expression: Now, we apply the chain rule. We multiply the derivative of the outer function by the derivative of the inner function. This gives us: f′(x)=−3×5(4x2−7x)4×(8x−7)f'(x) = -3 \times 5(4x^2 - 7x)^4 \times (8x - 7).
  6. Final Answer: Simplify the expression by multiplying the constants and keeping the rest of the expression in its current form. f′(x)=−15×(4x2−7x)4×(8x−7)f^{\prime}(x) = -15 \times (4x^2 - 7x)^4 \times (8x - 7).
  7. Final Answer: Simplify the expression by multiplying the constants and keeping the rest of the expression in its current form. f′(x)=−15×(4x2−7x)4×(8x−7)f'(x) = -15 \times (4x^2 - 7x)^4 \times (8x - 7).The final answer is f′(x)=−15(4x2−7x)4(8x−7)f'(x) = -15(4x^2 - 7x)^4(8x - 7).

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