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Derivative of f(x)=(3y^2+1)(2^y+1)

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/   2    \ / y    \
\3*y  + 1/*\2  + 1/

$$\left(2^{y} + 1\right) \left(3 y^{2} + 1\right)$$

(3*y^2 + 1)*(2^y + 1)

Detail solution

Apply the product rule:

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

$f{\left(y \right)} = 3 y^{2} + 1$ ; to find $\frac{d}{d y} f{\left(y \right)}$ :
1. Differentiate $3 y^{2} + 1$ 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: $y^{2}$ goes to $2 y$
    So, the result is: $6 y$
  2. The derivative of the constant $1$ is zero.
  The result is: $6 y$
$g{\left(y \right)} = 2^{y} + 1$ ; to find $\frac{d}{d y} g{\left(y \right)}$ :
1. Differentiate $2^{y} + 1$ term by term:
  1. $\frac{d}{d y} 2^{y} = 2^{y} \log{\left(2 \right)}$
  2. The derivative of the constant $1$ is zero.
  The result is: $2^{y} \log{\left(2 \right)}$
The result is: $2^{y} \left(3 y^{2} + 1\right) \log{\left(2 \right)} + 6 y \left(2^{y} + 1\right)$
Now simplify:

$2^{y} \left(3 y^{2} + 1\right) \log{\left(2 \right)} + 6 y \left(2^{y} + 1\right)$

The answer is:

$2^{y} \left(3 y^{2} + 1\right) \log{\left(2 \right)} + 6 y \left(2^{y} + 1\right)$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

    / y    \    y /   2    \       
6*y*\2  + 1/ + 2 *\3*y  + 1/*log(2)

$$2^{y} \left(3 y^{2} + 1\right) \log{\left(2 \right)} + 6 y \left(2^{y} + 1\right)$$

Simplify

The second derivative [src]

       y    y    2    /       2\         y       
6 + 6*2  + 2 *log (2)*\1 + 3*y / + 12*y*2 *log(2)

$$12 \cdot 2^{y} y \log{\left(2 \right)} + 2^{y} \left(3 y^{2} + 1\right) \log{\left(2 \right)}^{2} + 6 \cdot 2^{y} + 6$$

Simplify

The third derivative [src]

 y /        2    /       2\              \       
2 *\18 + log (2)*\1 + 3*y / + 18*y*log(2)/*log(2)

$$2^{y} \left(18 y \log{\left(2 \right)} + \left(3 y^{2} + 1\right) \log{\left(2 \right)}^{2} + 18\right) \log{\left(2 \right)}$$

Simplify