d tanx
1/ (cosx)^2
The expression d tanx represents the derivative of the tangent function with respect to x. To find this derivative, we can start with the definition of the tangent function:
tanx = sinx / cosx
Then, we can differentiate both sides of this equation with respect to x using the quotient rule for derivatives:
d/dx(tanx) = (cosx d/dx(sinx) – sinx d/dx(cosx)) / (cosx)^2
The derivative of sinx with respect to x is cosx, and the derivative of cosx with respect to x is -sinx. Substituting these values in the quotient rule formula, we get:
d/dx(tanx) = (cosx * cosx + sinx * sinx) / (cosx)^2
d/dx(tanx) = 1 / cos^2(x)
Therefore, the derivative of the tangent function with respect to x is 1/cos^2(x), which can also be written as sec^2(x) where sec(x) is the secant function.
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