Find the Derivative - d/dx sin(3-x)^2

Problem

d(sin(3−x))/d(x)

Solution

1. Identify the outer function and the inner function to apply the Chain Rule. The expression is of the form ƒ*(g(x)) where ƒ(u)=sin(u) and u=(3−x)2

2. Differentiate the outer function sin(u) with respect to u which gives cos(u)

3. Apply the Chain Rule by multiplying by the derivative of the inner function (3−x)2

d(sin(3−x))/d(x)=cos(3−x)⋅d(3−x)/d(x)

4. Differentiate the inner function (3−x)2 using the Power Rule and the Chain Rule again.

d(3−x)/d(x)=2*(3−x)(2−1)⋅d(3−x)/d(x)

5. Calculate the derivative of the innermost part (3−x) which is −1

d(3−x)/d(x)=2*(3−x)⋅(−1)

6. Simplify the expression by combining the terms.

d(3−x)/d(x)=−2*(3−x)

7. Substitute this result back into the main derivative expression.

d(sin(3−x))/d(x)=cos(3−x)⋅(−2*(3−x))

8. Rearrange the factors for the final form.

d(sin(3−x))/d(x)=−2*(3−x)*cos(3−x)

Final Answer

d(sin(3−x))/d(x)=−2*(3−x)*cos(3−x)

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