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Derivative of:
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Identical expressions
sin^ two *(three -x^ two)*(e^(5x))
sinus of squared multiply by (3 minus x squared ) multiply by (e to the power of (5x))
sinus of to the power of two multiply by (three minus x to the power of two) multiply by (e to the power of (5x))
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Similar expressions
sin^2*(3+x^2)*(e^(5x))

The derivative of the function/ sin^2*(3-x^2)*(e^(5x))

Derivative of sin^2(3-x^2)(e^(5x))

The solution

You have entered [src]

   2/     2\  5*x
sin \3 - x /*e

$$e^{5 x} \sin^{2}{\left(3 - x^{2} \right)}$$

d /   2/     2\  5*x\
--\sin \3 - x /*e   /
dx

$$\frac{d}{d x} e^{5 x} \sin^{2}{\left(3 - x^{2} \right)}$$

Transform

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = \sin^{2}{\left(3 - x^{2} \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = \sin{\left(3 - x^{2} \right)}$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(3 - x^{2} \right)}$ :
  1. Let $u = 3 - x^{2}$ .
  2. The derivative of sine is cosine:
    
    $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(3 - x^{2}\right)$ :
    1. Differentiate $3 - x^{2}$ term by term:
      1. The derivative of the constant $3$ is zero.
      2. The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x^{2}$ goes to $2 x$
        
        So, the result is: $- 2 x$
      The result is: $- 2 x$
    The result of the chain rule is:
    
    $- 2 x \cos{\left(x^{2} - 3 \right)}$
  The result of the chain rule is:
  
  $- 4 x \sin{\left(3 - x^{2} \right)} \cos{\left(x^{2} - 3 \right)}$
$g{\left(x \right)} = e^{5 x}$ ; 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: $- 4 x e^{5 x} \sin{\left(3 - x^{2} \right)} \cos{\left(x^{2} - 3 \right)} + 5 e^{5 x} \sin^{2}{\left(3 - x^{2} \right)}$
Now simplify:

$\left(2 x \sin{\left(2 x^{2} - 6 \right)} - \frac{5 \cos{\left(2 x^{2} - 6 \right)}}{2} + \frac{5}{2}\right) e^{5 x}$

The answer is:

$\left(2 x \sin{\left(2 x^{2} - 6 \right)} - \frac{5 \cos{\left(2 x^{2} - 6 \right)}}{2} + \frac{5}{2}\right) e^{5 x}$

The graph

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The first derivative [src]

     2/     2\  5*x          /      2\  5*x    /     2\
5*sin \3 - x /*e    - 4*x*cos\-3 + x /*e   *sin\3 - x /

$$- 4 x e^{5 x} \sin{\left(3 - x^{2} \right)} \cos{\left(x^{2} - 3 \right)} + 5 e^{5 x} \sin^{2}{\left(3 - x^{2} \right)}$$

Simplify

The second derivative [src]

/      2/      2\      2    2/      2\        /      2\    /      2\      2    2/      2\           /      2\    /      2\\  5*x
\25*sin \-3 + x / - 8*x *sin \-3 + x / + 4*cos\-3 + x /*sin\-3 + x / + 8*x *cos \-3 + x / + 40*x*cos\-3 + x /*sin\-3 + x //*e

$$\left(- 8 x^{2} \sin^{2}{\left(x^{2} - 3 \right)} + 8 x^{2} \cos^{2}{\left(x^{2} - 3 \right)} + 40 x \sin{\left(x^{2} - 3 \right)} \cos{\left(x^{2} - 3 \right)} + 25 \sin^{2}{\left(x^{2} - 3 \right)} + 4 \sin{\left(x^{2} - 3 \right)} \cos{\left(x^{2} - 3 \right)}\right) e^{5 x}$$

Simplify

The third derivative [src]

/       2/      2\        2    2/      2\       /       2/      2\        2/      2\      2    /      2\    /      2\\         /      2\    /      2\        2    2/      2\            /      2\    /      2\\  5*x
\125*sin \-3 + x / - 120*x *sin \-3 + x / - 8*x*\- 3*cos \-3 + x / + 3*sin \-3 + x / + 8*x *cos\-3 + x /*sin\-3 + x // + 60*cos\-3 + x /*sin\-3 + x / + 120*x *cos \-3 + x / + 300*x*cos\-3 + x /*sin\-3 + x //*e

$$\left(- 120 x^{2} \sin^{2}{\left(x^{2} - 3 \right)} + 120 x^{2} \cos^{2}{\left(x^{2} - 3 \right)} - 8 x \left(8 x^{2} \sin{\left(x^{2} - 3 \right)} \cos{\left(x^{2} - 3 \right)} + 3 \sin^{2}{\left(x^{2} - 3 \right)} - 3 \cos^{2}{\left(x^{2} - 3 \right)}\right) + 300 x \sin{\left(x^{2} - 3 \right)} \cos{\left(x^{2} - 3 \right)} + 125 \sin^{2}{\left(x^{2} - 3 \right)} + 60 \sin{\left(x^{2} - 3 \right)} \cos{\left(x^{2} - 3 \right)}\right) e^{5 x}$$

Simplify

The graph

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Derivative of sin^2(3-x^2)(e^(5x))

The graph:

Piecewise:

The solution

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Identical expressions

Similar expressions

Derivative of sin^2*(3-x^2)*(e^(5x))

The graph:

Piecewise:

The solution

Derivative of sin^2(3-x^2)(e^(5x))