## d/dx(tanx)

### sec²x

To find the derivative of tanx with respect to x, we can use the quotient rule. The quotient rule states that if we have a function in the form of f(x) = g(x)/h(x), then its derivative is given by:

f'(x) = (g'(x)h(x) – g(x)h'(x))/(h(x))^2

In this case, g(x) = sinx and h(x) = cosx.

Let’s find the derivatives of g(x) and h(x) first:

g'(x) = cosx (the derivative of sinx is cosx),

h'(x) = -sinx (the derivative of cosx is -sinx).

Now, we can substitute the values into the quotient rule formula:

f'(x) = (cosx * cosx – sinx * (-sinx))/(cosx)^2

= (cos^2 x + sin^2 x)/(cosx)^2

Using the trigonometric identity sin^2 x + cos^2 x = 1, we can simplify further:

f'(x) = 1/(cosx)^2

Therefore, the derivative of tanx with respect to x is 1/(cosx)^2.

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