Differentiate w.r.t. x
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\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{\cos(x)})
Use the definition of secant.
\frac{\cos(x)\frac{\mathrm{d}}{\mathrm{d}x}(1)-\frac{\mathrm{d}}{\mathrm{d}x}(\cos(x))}{\left(\cos(x)\right)^{2}}
For any two differentiable functions, the derivative of the quotient of two functions is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the denominator squared.
-\frac{-\sin(x)}{\left(\cos(x)\right)^{2}}
The derivative of the constant 1 is 0, and the derivative of cos(x) is −sin(x).
\frac{\sin(x)}{\left(\cos(x)\right)^{2}}
Simplify.
\frac{1}{\cos(x)}\times \left(\frac{\sin(x)}{\cos(x)}\right)
Rewrite the quotient as a product of two quotients.
\sec(x)\times \left(\frac{\sin(x)}{\cos(x)}\right)
Use the definition of secant.
\sec(x)\tan(x)
Use the definition of tangent.