Derivative of $e^{2 x}$

Q: What is the derivative of exp(2x)?

The derivative of e^2 x is 2 e^2 x.

The derivative of $e^{2 x}$ is $2 e^{2 x}$ .

f(x) =

\frac{d}{dx}\left[e^{2 x}\right] = 2 e^{2 x}

Type any function of x; the derivative is computed symbolically in your browser.

\frac{d}{dx}\left[e^{2 x}\right] = 2 e^{2 x}

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Graph of f(x) = exp(2x) and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = 14.778.

How to work it out

1
We differentiate $f(x) = e^{2 x}$ with respect to x. The rule that applies is the chain rule.
$f(x) = e^{2 x}$
2
Chain rule: differentiate the outer function, keep the inside, then multiply by the derivative of the inside. The inside is $g(x) = 2 x$ .
$g(x) = 2 x,\quad g'(x) = 2$
3
The outer function is $e^{u}$ ; its derivative evaluated at g(x), times g'(x):
$f'(x) = \left[e^{u}\right]_{u=g(x)} \cdot g'(x)$
4
Simplify:
$f'(x) = 2 e^{2 x}$
5
Differentiating once more gives the second derivative, which tells you about curvature:
$f''(x) = 4 e^{2 x}$

LaTeX source of the formulas on this page

\[ \frac{d}{dx}\left[e^{2 x}\right] = 2 e^{2 x} \]
\[ f(x) = e^{2 x} \]
\[ g(x) = 2 x,\quad g'(x) = 2 \]
\[ f'(x) = \left[e^{u}\right]_{u=g(x)} \cdot g'(x) \]
\[ f'(x) = 2 e^{2 x} \]
\[ f''(x) = 4 e^{2 x} \]

Values of f and f' for exp(2x)

x	f(x)	f'(x)
-2	0.018315639	0.036631278
-1	0.13533528	0.27067057
0.5	2.7182818	5.4365637
1	7.3890561	14.778112
2	54.59815	109.1963
3	403.42879	806.85759

Questions

What is the derivative of exp(2x)?

The derivative of $e^{2 x}$ is $2 e^{2 x}$ .

What is the second derivative of exp(2x)?

4exp(2x).

Which rule do you use for exp(2x)?

The chain rule.

What is the slope of exp(2x) at x = 1?

14.778112.

Derivative of e2xe^{2 x}e2x

How to work it out

Values of f and f' for exp(2x)

Questions

Related derivatives

Derivative of $e^{2 x}$