Using the product rule and other derivative rules prove that

Solution

Rewrite (tan(x) ) to sin(x) / cos(x) Now rewrite cos(x) as 1/sec(x) Thus giving you: sin(x) / [1/sec(x)] Which simplifies to sin(x)*sec(x) Then take the derivative using the product rule: First*(derivative of the second) + second*[derivative of the first] sin(x)*(sec(x)tan(x)) + sec(x)*(cos(x)) This simplifies to: sin^2(x) / cos^2(x) + cos(x) / cos(x) Which simplifies to: (sec^2(x) - 1) + 1 Which simplifies to sec^2(x)