Identify Composite Function: We need to find the derivative of the function sin(−2x+7) 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. The outer function in this case is sin(u), and the inner function is u=−2x+7.
Derivative of Outer Function: First, we find the derivative of the outer function with respect to its argument u. The derivative of sin(u) with respect to u is cos(u).
Derivative of Inner Function: Next, we find the derivative of the inner function −2x+7 with respect to x. The derivative of −2x with respect to x is −2, and the derivative of a constant like 7 is 0.
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 cos(−2x+7)⋅(−2).
Simplify Final Answer: Simplify the expression to get the final answer. The derivative of sin(−2x+7) with respect to x is −2⋅cos(−2x+7).
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