Find the Derivative - d/dx (10-cos(x))/(10+sin(x))

Problem

d()/d(x)(10−cos(x))/(10+sin(x))

Solution

1. Identify the quotient rule formula for a function u/v which is (vd(u)/d(x)−ud(v)/d(x))/(v2)

2. Assign the numerator u=10−cos(x) and the denominator v=10+sin(x)

3. Differentiate the numerator to find d(u)/d(x)=sin(x)

4. Differentiate the denominator to find d(v)/d(x)=cos(x)

5. Substitute these components into the quotient rule formula.

((10+sin(x))*(sin(x))−(10−cos(x))*(cos(x)))/((10+sin(x))2)

6. Distribute the terms in the numerator.

(10*sin(x)+sin2(x)−(10*cos(x)−cos2(x)))/((10+sin(x))2)

7. Simplify the numerator by removing parentheses and grouping trigonometric terms.

(10*sin(x)+sin2(x)−10*cos(x)+cos2(x))/((10+sin(x))2)

8. Apply the Pythagorean identity sin2(x)+cos2(x)=1

(10*sin(x)−10*cos(x)+1)/((10+sin(x))2)

Final Answer

d()/d(x)(10−cos(x))/(10+sin(x))=(10*sin(x)−10*cos(x)+1)/((10+sin(x))2)

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