Question

Find, from first principles,the derivative of e^tanx​

Answer 1

[(sec^2)(x)]. [e^tan(x)]

Or

[e^(tan(x))]/[(cos^2)(x)].

Explanation:

d/dx(e^tan(x))

From the low:

[(d/du)(e^u)]=[(d/du)(u)]. [ln(e)]. [e^u]

=[(sec^2)(x)]. [e^(tan(x))]. [ln(e)]

=[(sec^2)(x)]. [e^(tan(x))]

or

[e^(tan(x))]/[(cos^2)(x)].

Because

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