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Derivative of:
Derivative of 6
Derivative of x^7*e^x
Derivative of 1/(x+1)^2
Derivative of sin(4x)
Identical expressions
x^ seven *e^x
x to the power of 7 multiply by e to the power of x
x to the power of seven multiply by e to the power of x
x7*ex
x⁷*e^x
x^7e^x
x7ex

The derivative of the function/ x^7*e^x

Derivative of x^7*e^x

The solution

You have entered [src]

 7  x
x *E

$$e^{x} x^{7}$$

x^7*E^x

Detail solution

Apply the product rule:

$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$

$f{\left(x \right)} = x^{7}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $x^{7}$ goes to $7 x^{6}$
$g{\left(x \right)} = e^{x}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. The derivative of $e^{x}$ is itself.
The result is: $x^{7} e^{x} + 7 x^{6} e^{x}$
Now simplify:

$x^{6} \left(x + 7\right) e^{x}$

The answer is:

$x^{6} \left(x + 7\right) e^{x}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

 7  x      6  x
x *e  + 7*x *e

$$x^{7} e^{x} + 7 x^{6} e^{x}$$

Simplify

The second derivative [src]

 5 /      2       \  x
x *\42 + x  + 14*x/*e

$$x^{5} \left(x^{2} + 14 x + 42\right) e^{x}$$

Simplify

The third derivative [src]

 4 /       3       2        \  x
x *\210 + x  + 21*x  + 126*x/*e

$$x^{4} \left(x^{3} + 21 x^{2} + 126 x + 210\right) e^{x}$$

Simplify

The graph

Derivative of x^7*e^x