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Find 
(d)/(dx)(9cos(-x+1))
Answer:

Find ddx(9cos(x+1)) \frac{d}{d x}(9 \cos (-x+1)) \newlineAnswer:

Full solution

Q. Find ddx(9cos(x+1)) \frac{d}{d x}(9 \cos (-x+1)) \newlineAnswer:
  1. Identify Functions: We need to find the derivative of the function 9cos(x+1)9\cos(-x+1) with respect to xx. We will use the chain rule, which 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. Derivative of Outer Function: First, let's identify the outer function and the inner function. The outer function is 9cos(u)9\cos(u), where uu is the inner function x+1-x+1.
  3. Derivative of Inner Function: The derivative of the outer function with respect to uu (where u=x+1u = -x+1) is 9sin(u)-9\sin(u), because the derivative of cos(u)\cos(u) with respect to uu is sin(u)-\sin(u), and we have to multiply by the constant 99.
  4. Apply Chain Rule: Now we need to find the derivative of the inner function u=x+1u = -x+1 with respect to xx. The derivative of x-x with respect to xx is 1-1, and the derivative of a constant (like +1+1) is 00. So, the derivative of uu with respect to xx is 1-1.
  5. Final Derivative: Using the chain rule, we multiply the derivative of the outer function by the derivative of the inner function. This gives us 9sin(u)×(1)-9\sin(u) \times (-1).
  6. Final Derivative: Using the chain rule, we multiply the derivative of the outer function by the derivative of the inner function. This gives us 9sin(u)×(1)-9\sin(u) \times (-1).Substitute back the value of uu (x+1-x+1) into the derivative. So, the final derivative is 9sin(x+1)×(1)-9\sin(-x+1) \times (-1) which simplifies to 9sin(x+1)9\sin(-x+1).

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