Derivative Of Tan(X) Using Quotient Rule Of Differentiation And The Derivatives Of Sin(X) And Cos(X)

Sec^2

The derivative of tan(x) is found using the quotient rule of differentiation. Recall that

tan(x) = sin(x) / cos(x)

Then, applying the quotient rule, we get

d/dx [tan(x)] = [cos(x) d/dx(sin(x)) – sin(x) d/dx(cos(x))] / [cos^2(x)]

Now, we need to find the derivatives of sin(x) and cos(x) which are:
d/dx(sin(x)) = cos(x)

d/dx(cos(x)) = -sin(x)

Substituting these into the quotient rule formula we get;

d/dx [tan(x)] = [cos(x) cos(x) – sin(x)(-sin(x))] / [cos^2(x)]

= cos^2(x) + sin^2(x) / cos^2(x)

Since cos^2(x) + sin^2(x) = 1, then;

d/dx [tan(x)] = 1/cos^2(x)

Therefore, the derivative of tan(x) is 1/cos^2(x).

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Derivative Of Tan(X) Using Quotient Rule Of Differentiation And The Derivatives Of Sin(X) And Cos(X)