Derivative of tan(x) using the quotient rule, done in 1 minute!

In this video, we will find the derivative of the tangent function using the quotient rule. The derivative with respect to x of tan(x) is obtained by differentiating the numerator and denominator separately. We apply the quotient rule (low d high - high d low) / (low^2).

1. Let the numerator be sin(x) and the denominator be cos(x).
2. Apply the quotient rule (cos(x) * d/dx[sin(x)] - sin(x) * d/dx[cos(x)]) / (cos^2(x)).
3. Differentiate sin(x) to get cos(x), and differentiate cos(x) to get -sin(x).
4. Substitute the values (cos(x) * cos(x) - sin(x) * (-sin(x))) / (cos^2(x)).
5. Simplify (cos^2(x) + sin^2(x)) / (cos^2(x)).
6. Use the trigonometric identity sin^2(x) + cos^2(x) = 1.
7. Substitute the identity 1 / (cos^2(x)).
8. Recall that sec^2(x) = 1 / cos^2(x).
9. Final result sec^2(x).

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