Find the Derivative - d/dx (5-sec(x))/(tan(x))

Problem

d()/d(x)(5−sec(x))/tan(x)

Solution

1. Identify the rule needed for the derivative of a quotient, which is the quotient rule: d()/d(x)u/v=(vd(u)/d(x)−ud(v)/d(x))/(v2)

2. Assign the numerator and denominator functions: u=5−sec(x) and v=tan(x)

3. Differentiate the individual components: d(5−sec(x))/d(x)=−sec(x)*tan(x) and d(tan(x))/d(x)=sec2(x)

4. Substitute these into the quotient rule formula:

(tan(x)*(−sec(x)*tan(x))−(5−sec(x))*(sec2(x)))/tan2(x)

5. Expand the terms in the numerator:

(−sec(x)*tan2(x)−5*sec2(x)+sec3(x))/tan2(x)

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

(−sec(x)*(sec2(x)−1)−5*sec2(x)+sec3(x))/tan2(x)

7. Distribute and combine like terms:

(−sec3(x)+sec(x)−5*sec2(x)+sec3(x))/tan2(x)

8. Cancel the sec3(x) terms:

(sec(x)−5*sec2(x))/tan2(x)

9. Factor the numerator to reach the final simplified form:

(sec(x)*(1−5*sec(x)))/tan2(x)

Final Answer

d()/d(x)(5−sec(x))/tan(x)=(sec(x)−5*sec2(x))/tan2(x)

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