Derivatives of other trigonometric functions Verify the following derivative formulas using the Quotient Rule. ddx(sec⁡x)=sec⁡xtan⁡x

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Derivatives of other trigonometric functions Verify the following derivative formulas using the Quotient Rule.  ddx(sec⁡x)=sec⁡xtan⁡x

Philip Williams · Accepted Answer

Step 1  The secant function in terms of cosine function can be written as sec⁡x=1cos⁡x. Tangent function can be written in terms of sine, cosine function as tan⁡x=sin⁡xcos⁡x.  The quotient rule of differentiation states that ddx(f(x)g(x))=ddx(f(x))g(x)−f(x)ddx(g(x))(g(x))2. For the given problem f(x)=1 and g(x)=cos⁡x. The derivative of cos⁡x is −sin⁡x.  Step 2  Need to compute ddxsec⁡x. Use sec⁡x=1cos⁡x and then apply the quotient rule.  ddxsec⁡x=ddx(1cos⁡x)  =ddx(1)cos⁡x−1⋅ddxcos⁡x(cos⁡x)2  =0−(−sin⁡x)cos2x  =sin⁡xcos2x  =1cos⁡xsin⁡xcos⁡x  =sec⁡xtan⁡x  Hence, the given derivative formula is verified.

Derivatives of other trigonometric functions Verify the following derivative formulas

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