Find the Derivative Using Product Rule - d/dx sec(x)

Problem

d(sec(x))/d(x)

Solution

1. Identify the function as a product by rewriting the secant function in terms of sine and cosine.

sec(x)=1/cos(x)

2. Rewrite the expression to use the product rule by expressing the reciprocal as a product of 1 and (cos(x))(−1)

sec(x)=1⋅(cos(x))(−1)

3. Apply the product rule formula (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x) where u=1 and v=(cos(x))(−1)

d(sec(x))/d(x)=1⋅d(cos(x))/d(x)+(cos(x))(−1)⋅d(1)/d(x)

4. Differentiate the individual components using the chain rule for the first term and the constant rule for the second term.

d(sec(x))/d(x)=1⋅(−1*(cos(x))(−2)⋅(−sin(x)))+(cos(x))(−1)⋅0

5. Simplify the resulting expression by combining the terms and signs.

d(sec(x))/d(x)=sin(x)/cos2(x)

6. Separate the fraction into a product of two trigonometric ratios to reach the standard form.

d(sec(x))/d(x)=1/cos(x)⋅sin(x)/cos(x)

7. Substitute the trigonometric identities sec(x)=1/cos(x) and tan(x)=sin(x)/cos(x)

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

Final Answer

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

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