Mister Exam

Lang:

Other calculators

How to use it?
Derivative of:
Derivative of 3^(2*x)
Derivative of (x-3)^4
Derivative of ln(1+x)
Derivative of ln(1-x)
Identical expressions
e^(4sqrtx+x^ two)
e to the power of (4 square root of x plus x squared )
e to the power of (4 square root of x plus x to the power of two)
e^(4√x+x^2)
e(4sqrtx+x2)
e4sqrtx+x2
e^(4sqrtx+x²)
e to the power of (4sqrtx+x to the power of 2)
e^4sqrtx+x^2
Similar expressions
e^(4sqrtx-x^2)

The derivative of the function/ e^(4sqrtx+x^2)

Derivative of e^(4sqrtx+x^2)

The solution

You have entered [src]

     ___    2
 4*\/ x  + x 
E

$$e^{4 \sqrt{x} + x^{2}}$$

E^(4*sqrt(x) + x^2)

Detail solution

Let $u = 4 \sqrt{x} + x^{2}$ .
The derivative of $e^{u}$ is itself.
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(4 \sqrt{x} + x^{2}\right)$ :
1. Differentiate $4 \sqrt{x} + x^{2}$ term by term:
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$
    So, the result is: $\frac{2}{\sqrt{x}}$
  2. Apply the power rule: $x^{2}$ goes to $2 x$
  The result is: $2 x + \frac{2}{\sqrt{x}}$
The result of the chain rule is:

$\left(2 x + \frac{2}{\sqrt{x}}\right) e^{4 \sqrt{x} + x^{2}}$
Now simplify:

$\frac{2 \left(x^{\frac{3}{2}} + 1\right) e^{4 \sqrt{x} + x^{2}}}{\sqrt{x}}$

The answer is:

$\frac{2 \left(x^{\frac{3}{2}} + 1\right) e^{4 \sqrt{x} + x^{2}}}{\sqrt{x}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                   ___    2
/        2  \  4*\/ x  + x 
|2*x + -----|*e            
|        ___|              
\      \/ x /

$$\left(2 x + \frac{2}{\sqrt{x}}\right) e^{4 \sqrt{x} + x^{2}}$$

Simplify

The second derivative [src]

/                        2\   2       ___
|     1       /      1  \ |  x  + 4*\/ x 
|2 - ---- + 4*|x + -----| |*e            
|     3/2     |      ___| |              
\    x        \    \/ x / /

$$\left(4 \left(x + \frac{1}{\sqrt{x}}\right)^{2} + 2 - \frac{1}{x^{\frac{3}{2}}}\right) e^{4 \sqrt{x} + x^{2}}$$

Simplify

The third derivative [src]

/             3                                    \   2       ___
|  /      1  \      3        /     1  \ /      1  \|  x  + 4*\/ x 
|8*|x + -----|  + ------ + 6*|2 - ----|*|x + -----||*e            
|  |      ___|       5/2     |     3/2| |      ___||              
\  \    \/ x /    2*x        \    x   / \    \/ x //

$$\left(6 \left(2 - \frac{1}{x^{\frac{3}{2}}}\right) \left(x + \frac{1}{\sqrt{x}}\right) + 8 \left(x + \frac{1}{\sqrt{x}}\right)^{3} + \frac{3}{2 x^{\frac{5}{2}}}\right) e^{4 \sqrt{x} + x^{2}}$$

Simplify

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of e^(4sqrtx+x^2)

The graph:

Piecewise:

The solution