The derivative of tanx with respect to x can be found using the quotient rule of differentiation:
Let y = tanx
Then, we have:
y’ = d/dx (tanx)
Using the quotient rule, we get:
y’ = [sec^2x (d/dx)(x)] – [(tanx)(d/dx)(sec^2x)]
y’ = [sec^2x] – [(tanx)(2secx)(d/dx)(sinx)]
y’ = sec^2x – 2tanxsecxsinx
Therefore, the derivative of tanx with respect to x is sec^2x – 2tanxsecxsinx.
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