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

Problem

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

Solution

1. Identify the function as the reciprocal of the cosine function, which allows for the application of the chain rule or power rule.

sec(x)=(cos(x))(−1)

2. Apply the power rule and the chain rule by treating u=cos(x) as the inner function.

d(u(−1))/d(x)=−1*(u)(−2)⋅d(u)/d(x)

3. Substitute the inner function u=cos(x) and its derivative d(cos(x))/d(x)=−sin(x) into the expression.

d(sec(x))/d(x)=−(cos(x))(−2)⋅(−sin(x))

4. Simplify the signs and rewrite the expression using trigonometric identities.

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

5. Separate the fraction to express the result in terms of standard trigonometric functions.

d(sec(x))/d(x)=1/cos(x)⋅sin(x)/cos(x)

6. Recognize that 1/cos(x)=sec(x) and sin(x)/cos(x)=tan(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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