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Derivative of arctg(3x)
Derivative of y=(x^2+3)^4(2x^3-5)^3
Derivative of ln(1−cos(x))
Derivative of √(2x²-4x+1)
Identical expressions
y=(x^ two + three)^ four (2x^ three - five)^ three
y equally (x squared plus 3) to the power of 4(2x cubed minus 5) cubed
y equally (x to the power of two plus three) to the power of four (2x to the power of three minus five) to the power of three
y=(x2+3)4(2x3-5)3
y=x2+342x3-53
y=(x²+3)⁴(2x³-5)³
y=(x to the power of 2+3) to the power of 4(2x to the power of 3-5) to the power of 3
y=x^2+3^42x^3-5^3
Similar expressions
y=(x^2-3)^4(2x^3-5)^3
y=(x^2+3)^4(2x^3+5)^3

The derivative of the function/ y=(x^2+3)^4(2x^3-5)^3

Derivative of y=(x^2+3)^4(2x^3-5)^3

The solution

You have entered [src]

        4           3
/ 2    \  /   3    \ 
\x  + 3/ *\2*x  - 5/

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

  /        4           3\
d |/ 2    \  /   3    \ |
--\\x  + 3/ *\2*x  - 5/ /
dx

$$\frac{d}{d x} \left(x^{2} + 3\right)^{4} \left(2 x^{3} - 5\right)^{3}$$

Transform

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = \left(x^{2} + 3\right)^{4}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = x^{2} + 3$ .
2. Apply the power rule: $u^{4}$ goes to $4 u^{3}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x^{2} + 3\right)$ :
  1. Differentiate $x^{2} + 3$ term by term:
    1. Apply the power rule: $x^{2}$ goes to $2 x$
    2. The derivative of the constant $3$ is zero.
    The result is: $2 x$
  The result of the chain rule is:
  
  $8 x \left(x^{2} + 3\right)^{3}$
$g{\left(x \right)} = \left(2 x^{3} - 5\right)^{3}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = 2 x^{3} - 5$ .
2. Apply the power rule: $u^{3}$ goes to $3 u^{2}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(2 x^{3} - 5\right)$ :
  1. Differentiate $2 x^{3} - 5$ 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: $x^{3}$ goes to $3 x^{2}$
      So, the result is: $6 x^{2}$
    2. The derivative of the constant $\left(-1\right) 5$ is zero.
    The result is: $6 x^{2}$
  The result of the chain rule is:
  
  $18 x^{2} \left(2 x^{3} - 5\right)^{2}$
The result is: $18 x^{2} \left(x^{2} + 3\right)^{4} \left(2 x^{3} - 5\right)^{2} + 8 x \left(x^{2} + 3\right)^{3} \left(2 x^{3} - 5\right)^{3}$
Now simplify:

$2 x \left(x^{2} + 3\right)^{3} \left(2 x^{3} - 5\right)^{2} \cdot \left(8 x^{3} + 9 x \left(x^{2} + 3\right) - 20\right)$

The answer is:

$2 x \left(x^{2} + 3\right)^{3} \left(2 x^{3} - 5\right)^{2} \cdot \left(8 x^{3} + 9 x \left(x^{2} + 3\right) - 20\right)$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

            3           3                 4           2
    / 2    \  /   3    \        2 / 2    \  /   3    \ 
8*x*\x  + 3/ *\2*x  - 5/  + 18*x *\x  + 3/ *\2*x  - 5/

$$18 x^{2} \left(x^{2} + 3\right)^{4} \left(2 x^{3} - 5\right)^{2} + 8 x \left(x^{2} + 3\right)^{3} \left(2 x^{3} - 5\right)^{3}$$

Simplify

The second derivative [src]

          2             /             2                          2                                         \
  /     2\  /        3\ |  /        3\  /       2\       /     2\  /        3\       3 /        3\ /     2\|
4*\3 + x / *\-5 + 2*x /*\2*\-5 + 2*x / *\3 + 7*x / + 9*x*\3 + x / *\-5 + 8*x / + 72*x *\-5 + 2*x /*\3 + x //

$$4 \left(x^{2} + 3\right)^{2} \cdot \left(2 x^{3} - 5\right) \left(72 x^{3} \left(x^{2} + 3\right) \left(2 x^{3} - 5\right) + 9 x \left(x^{2} + 3\right)^{2} \cdot \left(8 x^{3} - 5\right) + 2 \cdot \left(7 x^{2} + 3\right) \left(2 x^{3} - 5\right)^{2}\right)$$

Simplify

The third derivative [src]

            /          3 /           2                            \                  3                               2                                     2                        \
   /     2\ |  /     2\  |/        3\        6       3 /        3\|       /        3\  /       2\       2 /        3\  /     2\ /       2\       2 /     2\  /        3\ /        3\|
12*\3 + x /*\3*\3 + x / *\\-5 + 2*x /  + 36*x  + 36*x *\-5 + 2*x // + 4*x*\-5 + 2*x / *\9 + 7*x / + 36*x *\-5 + 2*x / *\3 + x /*\3 + 7*x / + 72*x *\3 + x / *\-5 + 2*x /*\-5 + 8*x //

$$12 \left(x^{2} + 3\right) \left(72 x^{2} \left(x^{2} + 3\right)^{2} \cdot \left(2 x^{3} - 5\right) \left(8 x^{3} - 5\right) + 36 x^{2} \left(x^{2} + 3\right) \left(7 x^{2} + 3\right) \left(2 x^{3} - 5\right)^{2} + 4 x \left(7 x^{2} + 9\right) \left(2 x^{3} - 5\right)^{3} + 3 \left(x^{2} + 3\right)^{3} \cdot \left(36 x^{6} + 36 x^{3} \cdot \left(2 x^{3} - 5\right) + \left(2 x^{3} - 5\right)^{2}\right)\right)$$

Simplify

The graph

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Derivative of y=(x^2+3)^4(2x^3-5)^3

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The solution