Question

derivative of tan'x​

Answer 1

To find the derivative of tangent of x, we'll start by writing tan x as sin x/cos x and then use the quotient rule to differentiate. The quotient rule says that if two functions are differentiable, then the quotient is also differentiable. Here's the quotient rule applied to tan x when in form of sin x/cos x:

quotient_rule_derivative_tan_x

Now we know that the derivative of sin x is cos x and the derivative of cos x is -sin x. Substituting these derivatives in the parentheses and simplifying, we get:

simplify

Now there are two trigonometric identities we can use to simplify this problem:

sin2x + cos2x = 1

sec x = 1/cos x

the_result

And that's it, we are done! The derivative of tan x is sec2x.

However, there may be more to finding derivatives of tangent. In the general case, tan x is the tangent of a function of x, such as tan g(x). Note in the simple case, g(x) = x.

Generally, we are looking for:

general_form_for_derivative_of_tan_g(x)

For example:

g(x)=3x

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Answer 2

Once all those pieces are in place, the differentiation goes as follows:

d
d
x
tan
x

=
d
d
x
sin
x
cos
x


=
cos
x
.
cos
x
−
sin
x
.
(
−
sin
x
)
cos
2
x
(using Quotient Rule)

=
cos
2
x
+
sin
2
x=1cos2x(using the Pythagorean Identity)
=sec2x