Find the Derivative - d/dx y=(7-sec(x))/(tan(x))

Problem

d()/d(x)(7−sec(x))/tan(x)

Solution

1. Identify the rule needed for differentiation. Since the function is a quotient of two trigonometric functions, use the quotient rule: d()/d(x)u/v=(vd(u)/d(x)−ud(v)/d(x))/(v2)

2. Assign the numerator and denominator functions. Let u=7−sec(x) and v=tan(x)

3. Differentiate the individual components. The derivative of the numerator is d(7−sec(x))/d(x)=−sec(x)*tan(x) The derivative of the denominator is d(tan(x))/d(x)=sec2(x)

4. Substitute these values into the quotient rule formula.

d(y)/d(x)=(tan(x)*(−sec(x)*tan(x))−(7−sec(x))*(sec2(x)))/tan2(x)

5. Distribute and expand the terms in the numerator.

d(y)/d(x)=(−sec(x)*tan2(x)−7*sec2(x)+sec3(x))/tan2(x)

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

d(y)/d(x)=(−sec(x)*(sec2(x)−1)−7*sec2(x)+sec3(x))/tan2(x)

7. Combine like terms.

d(y)/d(x)=(−sec3(x)+sec(x)−7*sec2(x)+sec3(x))/tan2(x)

d(y)/d(x)=(sec(x)−7*sec2(x))/tan2(x)

8. Factor out sec(x) from the numerator.

d(y)/d(x)=(sec(x)*(1−7*sec(x)))/tan2(x)

Final Answer

d()/d(x)(7−sec(x))/tan(x)=(sec(x)−7*sec2(x))/tan2(x)

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