Discover How to Find the Derivative of Tan(x) with Respect to x Using the Quotient Rule and Get Expert Insights on Its Optimization.

d/dx tan x

sec^2 x

To find the derivative of tan(x) with respect to x, we use the quotient rule. Recall that the quotient rule states:

(f/g)’ = (f’g – fg’) / g^2

where f’ and g’ are the derivatives of f and g, respectively.

In this case, f = sin(x) and g = cos(x), so we have:

tan(x) = sin(x) / cos(x)

Taking the derivatives of f and g, we get:

f’ = cos(x) and g’ = -sin(x)

Using the quotient rule, we have:

(tan(x))’ = ((cos(x))(cos(x)) – (sin(x))(-sin(x))) / (cos(x))^2

Simplifying, we have:

(tan(x))’ = (cos^2(x) + sin^2(x)) / (cos^2(x))

Since cos^2(x) + sin^2(x) = 1, we have:

(tan(x))’ = 1 / cos^2(x)

And we know that:

sec^2(x) = 1 / cos^2(x)

So we can write:

(tan(x))’ = sec^2(x)

Therefore, the derivative of tan(x) with respect to x is sec^2(x).

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