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Derivative of:
Derivative of y=x^6*sin^2x
Derivative of y=x^6-7x
Derivative of y=(ctgx)sin2x
Derivative of cos(3t^2)
Identical expressions
y=x^ six *sin^2x
y equally x to the power of 6 multiply by sinus of squared x
y equally x to the power of six multiply by sinus of squared x
y=x6*sin2x
y=x⁶*sin²x
y=x to the power of 6*sin to the power of 2x
y=x^6sin^2x
y=x6sin2x

The derivative of the function/ y=x^6*sin^2x

Derivative of y=x^6*sin^2x

The solution

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 6    2   
x *sin (x)

$$x^{6} \sin^{2}{\left(x \right)}$$

x^6*sin(x)^2

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = x^{6}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $x^{6}$ goes to $6 x^{5}$
$g{\left(x \right)} = \sin^{2}{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = \sin{\left(x \right)}$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(x \right)}$ :
  1. The derivative of sine is cosine:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  The result of the chain rule is:
  
  $2 \sin{\left(x \right)} \cos{\left(x \right)}$
The result is: $2 x^{6} \sin{\left(x \right)} \cos{\left(x \right)} + 6 x^{5} \sin^{2}{\left(x \right)}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   5    2         6              
6*x *sin (x) + 2*x *cos(x)*sin(x)

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

Simplify

The second derivative [src]

   4 /      2       2 /   2         2   \                     \
2*x *\15*sin (x) - x *\sin (x) - cos (x)/ + 12*x*cos(x)*sin(x)/

$$2 x^{4} \left(- x^{2} \left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) + 12 x \sin{\left(x \right)} \cos{\left(x \right)} + 15 \sin^{2}{\left(x \right)}\right)$$

Simplify

The third derivative [src]

   3 /      2         2 /   2         2   \      3                                   \
4*x *\30*sin (x) - 9*x *\sin (x) - cos (x)/ - 2*x *cos(x)*sin(x) + 45*x*cos(x)*sin(x)/

$$4 x^{3} \left(- 2 x^{3} \sin{\left(x \right)} \cos{\left(x \right)} - 9 x^{2} \left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) + 45 x \sin{\left(x \right)} \cos{\left(x \right)} + 30 \sin^{2}{\left(x \right)}\right)$$

Simplify

The graph

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Identical expressions

Derivative of y=x^6*sin^2x

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The solution