d tan(x)
The derivative of the tangent function, d tan(x), is obtained by applying the chain rule of differentiation
The derivative of the tangent function, d tan(x), is obtained by applying the chain rule of differentiation. The chain rule states that if y = f(g(x)), then the derivative of y with respect to x is given by the product of the derivative of f(g(x)) with respect to g(x) and the derivative of g(x) with respect to x.
In the case of d tan(x), we can rewrite it as d tan(u), where u = x. Using the chain rule, we have:
d tan(u) = d tan(u)/du * du/dx
The derivative of tan(u) with respect to u can be found using the derivative formula for the tangent function. The derivative of tan(u) is sec^2(u). Therefore, we have:
d tan(u)/du = sec^2(u)
The derivative of u with respect to x is simply 1. Therefore, we have:
du/dx = 1
Substituting these values back into our chain rule equation, we get:
d tan(x) = sec^2(u) * 1
Simplifying this expression, we have:
d tan(x) = sec^2(x)
Therefore, the derivative of the tangent function, d tan(x), is equal to the secant squared function, sec^2(x).
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