Identify Composite Function: We are asked to find the derivative of the function −cos(−x+5) with respect to x. To do this, 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. The outer function in this case is −cos(u), where u=−x+5, and the inner function is u=−x+5.
Find Derivative of Outer Function: First, we find the derivative of the outer function −cos(u) with respect to u. The derivative of cos(u) with respect to u is −sin(u), so the derivative of −cos(u) with respect to u is sin(u).
Find Derivative of Inner Function: Next, we find the derivative of the inner function u=−x+5 with respect to x. The derivative of −x with respect to x is −1, and the derivative of a constant, like 5, is 0. So, the derivative of u with respect to x is −1.
Apply Chain Rule: Now, we apply the chain rule by multiplying the derivative of the outer function by the derivative of the inner function. This gives us sin(u)⋅(−1), where u=−x+5.
Substitute and Simplify: Substitute u=−x+5 into the expression sin(u)⋅(−1) to get the final derivative: sin(−x+5)⋅(−1).
Get Final Derivative: Simplify the expression to get the final answer. The final derivative is −sin(−x+5).
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