Derive the function (x^3cot(x^2)*2)/(1+e^(2x)) with respect to x

\frac{d}{dx}\left(\frac{2x^3\cdot \cot\left(x^2\right)}{1+e^{2x}}\right)

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Answer

$\frac{2\left(e^{2x}+1\right)\left(3x^{2}\cot\left(x^2\right)-2x^{4}\csc\left(x^2\right)^2\right)-4x^3\cot\left(x^2\right)e^{2x}}{\left(e^{2x}+1\right)^2}$

Step by step solution

Problem

$\frac{d}{dx}\left(\frac{2x^3\cdot \cot\left(x^2\right)}{1+e^{2x}}\right)$
1

Applying the quotient rule which states that if $f(x)$ and $g(x)$ are functions and $h(x)$ is the function defined by ${\displaystyle h(x) = \frac{f(x)}{g(x)}}$, where ${g(x) \neq 0}$, then ${\displaystyle h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}}$

$\frac{\left(e^{2x}+1\right)\frac{d}{dx}\left(2x^3\cot\left(x^2\right)\right)-2x^3\frac{d}{dx}\left(e^{2x}+1\right)\cot\left(x^2\right)}{\left(e^{2x}+1\right)^2}$
2

The derivative of a function multiplied by a constant is equal to the constant times the derivative of the function

$\frac{2\left(e^{2x}+1\right)\frac{d}{dx}\left(x^3\cot\left(x^2\right)\right)-2x^3\frac{d}{dx}\left(e^{2x}+1\right)\cot\left(x^2\right)}{\left(e^{2x}+1\right)^2}$
3

Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=x^3$ and $g=\cot\left(x^2\right)$

$\frac{2\left(e^{2x}+1\right)\left(x^3\frac{d}{dx}\left(\cot\left(x^2\right)\right)+\cot\left(x^2\right)\frac{d}{dx}\left(x^3\right)\right)-2x^3\frac{d}{dx}\left(e^{2x}+1\right)\cot\left(x^2\right)}{\left(e^{2x}+1\right)^2}$
4

The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

$\frac{2\left(e^{2x}+1\right)\left(x^3\frac{d}{dx}\left(\cot\left(x^2\right)\right)+3x^{2}\cot\left(x^2\right)\right)-2x^3\frac{d}{dx}\left(e^{2x}+1\right)\cot\left(x^2\right)}{\left(e^{2x}+1\right)^2}$
5

Taking the derivative of cotangent

$\frac{2\left(e^{2x}+1\right)\left(3x^{2}\cot\left(x^2\right)-x^3\frac{d}{dx}\left(x^2\right)\csc\left(x^2\right)^2\right)-2x^3\frac{d}{dx}\left(e^{2x}+1\right)\cot\left(x^2\right)}{\left(e^{2x}+1\right)^2}$
6

The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

$\frac{2\left(e^{2x}+1\right)\left(3x^{2}\cot\left(x^2\right)-1\cdot 2xx^3\csc\left(x^2\right)^2\right)-2x^3\frac{d}{dx}\left(e^{2x}+1\right)\cot\left(x^2\right)}{\left(e^{2x}+1\right)^2}$
7

The derivative of a sum of two functions is the sum of the derivatives of each function

$\frac{2\left(e^{2x}+1\right)\left(3x^{2}\cot\left(x^2\right)-1\cdot 2xx^3\csc\left(x^2\right)^2\right)-2x^3\left(\frac{d}{dx}\left(e^{2x}\right)+\frac{d}{dx}\left(1\right)\right)\cot\left(x^2\right)}{\left(e^{2x}+1\right)^2}$
8

The derivative of the constant function is equal to zero

$\frac{2\left(e^{2x}+1\right)\left(3x^{2}\cot\left(x^2\right)-1\cdot 2xx^3\csc\left(x^2\right)^2\right)-2x^3\left(\frac{d}{dx}\left(e^{2x}\right)+0\right)\cot\left(x^2\right)}{\left(e^{2x}+1\right)^2}$
9

Applying the derivative of the exponential function

$\frac{2\left(e^{2x}+1\right)\left(3x^{2}\cot\left(x^2\right)-1\cdot 2xx^3\csc\left(x^2\right)^2\right)-2x^3\left(1\frac{d}{dx}\left(2x\right)e^{2x}+0\right)\cot\left(x^2\right)}{\left(e^{2x}+1\right)^2}$
10

The derivative of a function multiplied by a constant is equal to the constant times the derivative of the function

$\frac{2\left(e^{2x}+1\right)\left(3x^{2}\cot\left(x^2\right)-1\cdot 2xx^3\csc\left(x^2\right)^2\right)-2x^3\left(1\cdot 2e^{2x}\cdot\frac{d}{dx}\left(x\right)+0\right)\cot\left(x^2\right)}{\left(e^{2x}+1\right)^2}$
11

The derivative of the linear function is equal to $1$

$\frac{2\left(e^{2x}+1\right)\left(3x^{2}\cot\left(x^2\right)-1\cdot 2xx^3\csc\left(x^2\right)^2\right)-2x^3\left(1\cdot 1\cdot 2e^{2x}+0\right)\cot\left(x^2\right)}{\left(e^{2x}+1\right)^2}$
12

Multiply $2$ times $-1$

$\frac{2\left(e^{2x}+1\right)\left(3x^{2}\cot\left(x^2\right)-2xx^3\csc\left(x^2\right)^2\right)-2x^3\left(2e^{2x}+0\right)\cot\left(x^2\right)}{\left(e^{2x}+1\right)^2}$
13

$x+0=x$, where $x$ is any expression

$\frac{2\left(e^{2x}+1\right)\left(3x^{2}\cot\left(x^2\right)-2xx^3\csc\left(x^2\right)^2\right)-2\cdot 2x^3\cot\left(x^2\right)e^{2x}}{\left(e^{2x}+1\right)^2}$
14

Multiply $2$ times $-2$

$\frac{2\left(e^{2x}+1\right)\left(3x^{2}\cot\left(x^2\right)-2xx^3\csc\left(x^2\right)^2\right)-4x^3\cot\left(x^2\right)e^{2x}}{\left(e^{2x}+1\right)^2}$
15

When multiplying exponents with same base you can add the exponents

$\frac{2\left(e^{2x}+1\right)\left(3x^{2}\cot\left(x^2\right)-2x^{4}\csc\left(x^2\right)^2\right)-4x^3\cot\left(x^2\right)e^{2x}}{\left(e^{2x}+1\right)^2}$

Answer

$\frac{2\left(e^{2x}+1\right)\left(3x^{2}\cot\left(x^2\right)-2x^{4}\csc\left(x^2\right)^2\right)-4x^3\cot\left(x^2\right)e^{2x}}{\left(e^{2x}+1\right)^2}$