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

Problem

d()/d(x)*sec(x)*tan(x)

Solution

1. Identify the rule needed for differentiation. Since the expression is a product of two functions, sec(x) and tan(x) use the product rule: (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x)

2. Assign the functions for the product rule. Let u=sec(x) and v=tan(x)

3. Differentiate each component. The derivative of sec(x) is sec(x)*tan(x) and the derivative of tan(x) is sec2(x)

4. Apply the product rule formula.

(d(sec(x))*tan(x))/d(x)=sec(x)d(tan(x))/d(x)+tan(x)d(sec(x))/d(x)

5. Substitute the derivatives into the expression.

(d(sec(x))*tan(x))/d(x)=sec(x)*sec2(x)+tan(x)*(sec(x)*tan(x))

6. Simplify the terms by combining powers and factoring.

(d(sec(x))*tan(x))/d(x)=sec3(x)+sec(x)*tan2(x)

7. Factor out the common term sec(x)

(d(sec(x))*tan(x))/d(x)=sec(x)*(sec2(x)+tan2(x))

Final Answer

(d(sec(x))*tan(x))/d(x)=sec(x)*(sec2(x)+tan2(x))

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