d/dx tanx
sec^2(x)
The derivative of tanx with respect to x can be found by applying the quotient rule.
Recall that the quotient rule states:
If y = f(x)/g(x), then y’ = (f'(x)g(x) – f(x)g'(x))/[g(x)]^2
Let f(x) = sinx and g(x) = cosx, then
tanx = sinx/cosx
Now, applying the quotient rule:
[tanx]’ = [(sinx)’cosx – sinx(cosx)’]/(cosx)^2
= [cosx*cosx + sinx*sinx]/(cosx)^2
Recall that sin²x + cos²x = 1, so we can simplify:
[tanx]’ = 1/(cosx)^2
Therefore, the derivative of tanx with respect to x is 1/(cosx)^2.
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