Identify Functions: We need to find the derivative of the function 9cos(−x+1) with respect to x. 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.
Derivative of Outer Function: First, let's identify the outer function and the inner function. The outer function is 9cos(u), where u is the inner function −x+1.
Derivative of Inner Function: The derivative of the outer function with respect to u (where u=−x+1) is −9sin(u), because the derivative of cos(u) with respect to u is −sin(u), and we have to multiply by the constant 9.
Apply Chain Rule: Now we need to find the derivative of the inner function u=−x+1 with respect to x. The derivative of −x with respect to x is −1, and the derivative of a constant (like +1) is 0. So, the derivative of u with respect to x is −1.
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).
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).Substitute back the value of u (−x+1) into the derivative. So, the final derivative is −9sin(−x+1)×(−1) which simplifies to 9sin(−x+1).
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