Derivative of tan(x)sec(x) | The Product Rule in Differentiation Explained

tan(x)sec(x) is the derivative of?

To find the derivative of tan(x)sec(x), we can use the product rule of differentiation

To find the derivative of tan(x)sec(x), we can use the product rule of differentiation.

First, let’s rewrite the expression as the product of tan(x) and sec(x): tan(x)sec(x) = tan(x) * sec(x).

Now, let’s differentiate each term separately.
The derivative of tan(x) can be found using the quotient rule: d/dx(tan(x)) = sec^2(x).

The derivative of sec(x) can be found using the chain rule: d/dx(sec(x)) = sec(x) * tan(x).

Now, let’s use the product rule:
d/dx(tan(x) * sec(x)) = tan(x) * d/dx(sec(x)) + sec(x) * d/dx(tan(x))

Plugging in the derivatives we found earlier:
= tan(x) * (sec(x) * tan(x)) + sec(x) * (sec^2(x))

Simplifying the expression:
= tan^2(x)sec(x) + sec^3(x)

Therefore, the derivative of tan(x)sec(x) is tan^2(x)sec(x) + sec^3(x).

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