Derivative of $\sqrt{x} e^{x}$

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

The derivative of √(x) e^x is e^x2 √(x) + √(x) e^x.

The derivative of $\sqrt{x} e^{x}$ is $\frac{e^{x}}{2 \sqrt{x}} + \sqrt{x} e^{x}$ .

f(x) =

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

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

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

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Graph of f(x) = sqrt(x) * exp(x) and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = 4.0774.

How to work it out

1
We differentiate $f(x) = \sqrt{x} e^{x}$ with respect to x. The rule that applies is the product rule.
$f(x) = \sqrt{x} e^{x}$
2
Product rule: $(uv)' = u'v + uv'$ . Let $u = \sqrt{x}$ and $v = e^{x}$ .
$u = \sqrt{x},\quad v = e^{x}$
3
Differentiate each piece:
$u' = \frac{1}{2 \sqrt{x}},\quad v' = e^{x}$
4
Put them in the rule:
$f'(x) = \left(\frac{1}{2 \sqrt{x}}\right)\left(e^{x}\right) + \left(\sqrt{x}\right)\left(e^{x}\right)$
5
Simplify:
$f'(x) = \frac{e^{x}}{2 \sqrt{x}} + \sqrt{x} e^{x}$
6
Differentiating once more gives the second derivative, which tells you about curvature:
$f''(x) = \frac{e^{x} 2 \sqrt{x} - e^{x} 2 \frac{1}{2 \sqrt{x}}}{\left(2 \sqrt{x}\right)^{2}} + \frac{e^{x}}{2 \sqrt{x}} + \sqrt{x} e^{x}$

LaTeX source of the formulas on this page

\[ \frac{d}{dx}\left[\sqrt{x} e^{x}\right] = \frac{e^{x}}{2 \sqrt{x}} + \sqrt{x} e^{x} \]
\[ f(x) = \sqrt{x} e^{x} \]
\[ u = \sqrt{x},\quad v = e^{x} \]
\[ u' = \frac{1}{2 \sqrt{x}},\quad v' = e^{x} \]
\[ f'(x) = \left(\frac{1}{2 \sqrt{x}}\right)\left(e^{x}\right) + \left(\sqrt{x}\right)\left(e^{x}\right) \]
\[ f'(x) = \frac{e^{x}}{2 \sqrt{x}} + \sqrt{x} e^{x} \]
\[ f''(x) = \frac{e^{x} 2 \sqrt{x} - e^{x} 2 \frac{1}{2 \sqrt{x}}}{\left(2 \sqrt{x}\right)^{2}} + \frac{e^{x}}{2 \sqrt{x}} + \sqrt{x} e^{x} \]

Values of f and f' for sqrt(x) * exp(x)

x	f(x)	f'(x)
0.5	1.165822	2.331644
1	2.7182818	4.0774227
2	10.449703	13.062129
3	34.78917	40.587366

Questions

What is the derivative of sqrt(x) * exp(x)?

The derivative of $\sqrt{x} e^{x}$ is $\frac{e^{x}}{2 \sqrt{x}} + \sqrt{x} e^{x}$ .

What is the second derivative of sqrt(x) * exp(x)?

(exp(x) * 2sqrt(x) - exp(x) * 2 * 1 / (2sqrt(x))) / (2sqrt(x))^2 + exp(x) / (2sqrt(x)) + sqrt(x) * exp(x).

Which rule do you use for sqrt(x) * exp(x)?

The product rule.

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

4.0774227.

Derivative of xex\sqrt{x} e^{x}x​ex

How to work it out

Values of f and f' for sqrt(x) * exp(x)

Questions

Related derivatives

Derivative of $\sqrt{x} e^{x}$