1
Solved example of derivatives of hyperbolic trigonometric functions
$\frac{d}{dx}\left(csch^2\left(4x^3+1\right)\right)$
The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$
$2csch\left(4x^3+1\right)^{\left(2-1\right)}\frac{d}{dx}\left(csch\left(4x^3+1\right)\right)$
Subtract the values $2$ and $-1$
$2csch\left(4x^3+1\right)\frac{d}{dx}\left(csch\left(4x^3+1\right)\right)$
2
The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$
$2csch\left(4x^3+1\right)\frac{d}{dx}\left(csch\left(4x^3+1\right)\right)$
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Taking the derivative of hyperbolic cosecant
$2\left(-1\right)csch\left(4x^3+1\right)csch\left(4x^3+1\right)coth\left(4x^3+1\right)\frac{d}{dx}\left(4x^3+1\right)$
$-2csch\left(4x^3+1\right)^2coth\left(4x^3+1\right)\frac{d}{dx}\left(4x^3+1\right)$
3
Taking the derivative of hyperbolic cosecant
$-2csch\left(4x^3+1\right)^2coth\left(4x^3+1\right)\frac{d}{dx}\left(4x^3+1\right)$
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4
The derivative of a sum of two or more functions is the sum of the derivatives of each function
$-2csch\left(4x^3+1\right)^2coth\left(4x^3+1\right)\left(\frac{d}{dx}\left(4x^3\right)+\frac{d}{dx}\left(1\right)\right)$
The derivative of the constant function ($1$) is equal to zero
$-2csch\left(4x^3+1\right)^2coth\left(4x^3+1\right)\left(\frac{d}{dx}\left(4x^3\right)+0\right)$
$x+0=x$, where $x$ is any expression
$-2csch\left(4x^3+1\right)^2coth\left(4x^3+1\right)\frac{d}{dx}\left(4x^3\right)$
5
The derivative of the constant function ($1$) is equal to zero
$-2csch\left(4x^3+1\right)^2coth\left(4x^3+1\right)\frac{d}{dx}\left(4x^3\right)$
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The derivative of a function multiplied by a constant ($4$) is equal to the constant times the derivative of the function
$-2\cdot 4csch\left(4x^3+1\right)^2coth\left(4x^3+1\right)\frac{d}{dx}\left(x^3\right)$
$-8csch\left(4x^3+1\right)^2coth\left(4x^3+1\right)\frac{d}{dx}\left(x^3\right)$
6
The derivative of a function multiplied by a constant ($4$) is equal to the constant times the derivative of the function
$-8csch\left(4x^3+1\right)^2coth\left(4x^3+1\right)\frac{d}{dx}\left(x^3\right)$
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The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$
$-8\cdot 3csch\left(4x^3+1\right)^2x^{\left(3-1\right)}coth\left(4x^3+1\right)$
Subtract the values $3$ and $-1$
$-24csch\left(4x^3+1\right)^2x^{2}coth\left(4x^3+1\right)$
7
The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$
$-24csch\left(4x^3+1\right)^2x^{2}coth\left(4x^3+1\right)$
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Final Answer
$-24csch\left(4x^3+1\right)^2x^{2}coth\left(4x^3+1\right)$