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Derivative of:
Derivative of cos(x/3)
Derivative of √x
Derivative of xcos(x)
Derivative of 4/(x^2)
Identical expressions
seven *x*e^(five *x)
7 multiply by x multiply by e to the power of (5 multiply by x)
seven multiply by x multiply by e to the power of (five multiply by x)
7*x*e(5*x)
7*x*e5*x
7xe^(5x)
7xe(5x)
7xe5x
7xe^5x

The derivative of the function/ 7*x*e^(5*x)

Derivative of 7xe^(5*x)

The solution

You have entered [src]

     5*x
7*x*e

$$7 x e^{5 x}$$

d /     5*x\
--\7*x*e   /
dx

$$\frac{d}{d x} 7 x e^{5 x}$$

Transform

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Apply the power rule: $x$ goes to $1$
  $g{\left(x \right)} = e^{5 x}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Let $u = 5 x$ .
  2. The derivative of $e^{u}$ is itself.
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 5 x$ :
    1. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x$ goes to $1$
      So, the result is: $5$
    The result of the chain rule is:
    
    $5 e^{5 x}$
  The result is: $5 x e^{5 x} + e^{5 x}$
So, the result is: $35 x e^{5 x} + 7 e^{5 x}$
Now simplify:

$\left(35 x + 7\right) e^{5 x}$

The answer is:

$\left(35 x + 7\right) e^{5 x}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   5*x         5*x
7*e    + 35*x*e

$$35 x e^{5 x} + 7 e^{5 x}$$

Simplify

The second derivative [src]

              5*x
35*(2 + 5*x)*e

$$35 \cdot \left(5 x + 2\right) e^{5 x}$$

Simplify

The third derivative [src]

               5*x
175*(3 + 5*x)*e

$$175 \cdot \left(5 x + 3\right) e^{5 x}$$

Simplify

The graph

Derivative of 7*x*e^(5*x)