derivative of tanx
sec^2x
The derivative of tanx can be found using the quotient rule of differentiation. Recall that the quotient rule states:
If y = u/v, where u and v are functions of x, then
dy/dx = (v*du/dx – u*dv/dx) / v^2
In this case, we can let u = sinx and v = cosx, since tanx = sinx/cosx. Then, using the chain rule, we have:
du/dx = cosx and dv/dx = -sinx
Substituting these values into the quotient rule formula gives:
dy/dx = (cosx*cosx – sinx*(-sinx)) / cos^2x
= (cos^2x + sin^2x) / cos^2x (since sin^2x + cos^2x = 1)
= 1/cos^2x
Therefore, the derivative of tanx is:
d/dx(tanx) = sec^2x
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