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Given the function f(x)=-5(5x^(2)+9x)^(6), find f^(')(x) in any form. Answer: f^(')(x)=

Given the function f(x)=−5(5x2+9x)6 f(x)=-5\left(5 x^{2}+9 x\right)^{6} , 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)=−5(5x2+9x)6 f(x)=-5\left(5 x^{2}+9 x\right)^{6} , 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)=−5(5x2+9x)6f(x)=-5(5x^{2}+9x)^{6}, 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 u6u^6 where uu is the inner function, and the inner function is 5x2+9x5x^2 + 9x.
  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 u6u^6 with respect to uu is 6u56u^5.
  4. Apply Chain Rule: Next, we will find the derivative of the inner function 5x2+9x5x^2 + 9x with respect to xx. The derivative of 5x25x^2 is 10x10x, and the derivative of 9x9x is 99. So, the derivative of the inner function is 10x+910x + 9.
  5. Simplify Expression: Now, we apply the chain rule by multiplying the derivative of the outer function by the derivative of the inner function. This gives us: f′(x)=−5×6(5x2+9x)5×(10x+9)f'(x) = -5 \times 6(5x^2 + 9x)^5 \times (10x + 9).
  6. Simplify Expression: Now, we apply the chain rule by multiplying the derivative of the outer function by the derivative of the inner function. This gives us: f′(x)=−5×6(5x2+9x)5×(10x+9)f'(x) = -5 \times 6(5x^2 + 9x)^5 \times (10x + 9). Simplify the expression by combining constants and applying the power rule. f′(x)=−30(5x2+9x)5×(10x+9)f'(x) = -30(5x^2 + 9x)^5 \times (10x + 9).

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