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y=(x²+3x)⁴
The derivative of the function/ y=(x²+3x)⁴

Derivative of y=(x²+3x)⁴

Function f() - derivative -N order at the point
v

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The solution

You have entered [src] 
          4
/ 2      \ 
\x  + 3*x/
(x2+3x)4\left(x^{2} + 3 x\right)^{4} 
  /          4\
d |/ 2      \ |
--\\x  + 3*x/ /
dx
ddx(x2+3x)4\frac{d}{d x} \left(x^{2} + 3 x\right)^{4}
Transform
Detail solution

  1. Let u=x2+3xu = x^{2} + 3 x.

  2. Apply the power rule: u4u^{4} goes to 4u34 u^{3}

  3. Then, apply the chain rule. Multiply by ddx(x2+3x)\frac{d}{d x} \left(x^{2} + 3 x\right):

    1. Differentiate x2+3xx^{2} + 3 x term by term:

      1. Apply the power rule: x2x^{2} goes to 2x2 x

      2. The derivative of a constant times a function is the constant times the derivative of the function.

        1. Apply the power rule: xx goes to 11

        So, the result is: 33

      The result is: 2x+32 x + 3

    The result of the chain rule is:

    4⋅(2x+3)(x2+3x)34 \cdot \left(2 x + 3\right) \left(x^{2} + 3 x\right)^{3}

  4. Now simplify:

    x3(x+3)3⋅(8x+12)x^{3} \left(x + 3\right)^{3} \cdot \left(8 x + 12\right)

The answer is:

x3(x+3)3⋅(8x+12)x^{3} \left(x + 3\right)^{3} \cdot \left(8 x + 12\right)

The graph
02468-8-6-4-2-1010-500000000500000000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
          3           
/ 2      \            
\x  + 3*x/ *(12 + 8*x)
(8x+12)(x2+3x)3\left(8 x + 12\right) \left(x^{2} + 3 x\right)^{3}
Simplify
The second derivative [src] 
   2        2 /           2              \
4*x *(3 + x) *\3*(3 + 2*x)  + 2*x*(3 + x)/
4x2(x+3)2⋅(2x(x+3)+3(2x+3)2)4 x^{2} \left(x + 3\right)^{2} \cdot \left(2 x \left(x + 3\right) + 3 \left(2 x + 3\right)^{2}\right)
Simplify
The third derivative [src] 
                       /         2              \
24*x*(3 + x)*(3 + 2*x)*\(3 + 2*x)  + 3*x*(3 + x)/
24x(x+3)(2x+3)(3x(x+3)+(2x+3)2)24 x \left(x + 3\right) \left(2 x + 3\right) \left(3 x \left(x + 3\right) + \left(2 x + 3\right)^{2}\right)
Simplify
The graph
Derivative of y=(x²+3x)⁴
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