Find the Derivative - d/dx (4-sec(x))/(tan(x))

Problem

d()/d(x)(4−sec(x))/tan(x)

Solution

1. Identify the quotient rule for differentiation, which states that for a function y=u/v the derivative is (vd(u)/d(x)−ud(v)/d(x))/(v2)

2. Assign the numerator u=4−sec(x) and the denominator v=tan(x)

3. Differentiate the numerator to find d(u)/d(x)=−sec(x)*tan(x)

4. Differentiate the denominator to find d(v)/d(x)=sec2(x)

5. Substitute these components into the quotient rule formula:

(tan(x)*(−sec(x)*tan(x))−(4−sec(x))*(sec2(x)))/tan2(x)

6. Distribute the terms in the numerator:

(−sec(x)*tan2(x)−4*sec2(x)+sec3(x))/tan2(x)

7. Simplify the expression using the trigonometric identity tan2(x)=sec2(x)−1

(−sec(x)*(sec2(x)−1)−4*sec2(x)+sec3(x))/tan2(x)

8. Expand and combine like terms:

(−sec3(x)+sec(x)−4*sec2(x)+sec3(x))/tan2(x)

9. Reduce the numerator:

(sec(x)−4*sec2(x))/tan2(x)

10. Rewrite in terms of sine and cosine to simplify further:

(1/cos(x)−4/cos2(x))/sin2(x)/cos2(x)

11. Multiply the numerator and denominator by cos2(x)

(cos(x)−4)/sin2(x)

Final Answer

d()/d(x)(4−sec(x))/tan(x)=(cos(x)−4)/sin2(x)

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