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Find the derivative of y = e^ x

April 29, 2026Mathematics

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Answer

e^{\tan x} \sec^2 x

Graph of y = e^x

Key points of y = e^x
Point	x	y
y-intercept	0	1

5 steps

To find the derivative of $y = e^{\tan x}$ , we use the chain rule.

Identify the outer and inner functions.
Let $u = \tan x$ .
Then $y = e^u$ .
Differentiate $y$ with respect to $u$ .
$\frac{dy}{du} = \frac{d}{du}(e^u) = e^u$
Differentiate $u$ with respect to $x$ .
$\frac{du}{dx} = \frac{d}{dx}(\tan x) = \sec^2 x$
Apply the chain rule, which states $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$ .
$\frac{dy}{dx} = e^u \cdot \sec^2 x$
Substitute $u = \tan x$ back into the expression.
$\frac{dy}{dx} = e^{\tan x} \cdot \sec^2 x$
The final answer is $e^{\tan x} \sec^2 x$ .

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