Differentiation of e^tanx | Differentiate e^tanx

The differentiation of e^tanx is equal to sec²x e^tanx. In this post, we will learn how to differentiate e to the power tanx with respect to x.

The following two formulas will be used to find the derivative of e^tanx.

1. d/dx(tanx) = sec²x.
2. log_a a^k =k.

Now, we will learn to find the derivative of e to the power tanx with respect to x.

How to Find the derivative of e^tanx

Question: How to differentiate e^tanx?

Solution:

Let y= e^tanx

Taking natural logarithms on both sides, we get that (natural logarithm means the logarithm with base e, i.e. log_e)

log y = tanx (here we used the logarithm rule log_ee^k =k.)

Now, differentiate both sides with respect to x. This will give us

$\dfrac{1}{y} \dfrac{dy}{dx}$ = sec²x as we know that the derivative of tanx is sec²x.

⇒ $\dfrac{dy}{dx}$ = y sec²x

Putting the value of y, that is, y=e^tanx, we get from above that

$\dfrac{d}{dx}$(e^tanx) = sec²x e^tanx.

Thus, the differentiation of e^tanx with respect to x is equal to e^tanx sec²x.

Video solution on derivative of e^tanx:

ALSO READ:

Derivative of arc(cotx): The derivative of arc(cotx) is -1/(1+x²).

Differentiate e^sinx: The derivative of e^sinx is e^sinx cosx.

Differentiate e^cosx: The derivative of e^cosx is -e^cosx sinx.

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