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Derivatives of hyperbolic trigonometric functions Calculator

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1

Solved example of derivatives of hyperbolic trigonometric functions

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

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

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

Subtract the values $2$ and $-1$

$2\mathrm{csch}\left(4x^3+1\right)^{1}\frac{d}{dx}\left(\mathrm{csch}\left(4x^3+1\right)\right)$
4

Any expression to the power of $1$ is equal to that same expression

$2\mathrm{csch}\left(4x^3+1\right)\frac{d}{dx}\left(\mathrm{csch}\left(4x^3+1\right)\right)$
5

Taking the derivative of hyperbolic cosecant

$2\left(-1\right)\mathrm{csch}\left(4x^3+1\right)\mathrm{csch}\left(4x^3+1\right)\mathrm{coth}\left(4x^3+1\right)\frac{d}{dx}\left(4x^3+1\right)$
6

Multiply $2$ times $-1$

$-2\mathrm{csch}\left(4x^3+1\right)\mathrm{csch}\left(4x^3+1\right)\mathrm{coth}\left(4x^3+1\right)\frac{d}{dx}\left(4x^3+1\right)$
7

When multiplying two powers that have the same base ($\mathrm{csch}\left(4x^3+1\right)$), you can add the exponents

$-2\mathrm{csch}\left(4x^3+1\right)^2\mathrm{coth}\left(4x^3+1\right)\frac{d}{dx}\left(4x^3+1\right)$
8

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

$-2\mathrm{csch}\left(4x^3+1\right)^2\mathrm{coth}\left(4x^3+1\right)\left(\frac{d}{dx}\left(4x^3\right)+\frac{d}{dx}\left(1\right)\right)$

$-2\mathrm{csch}\left(4x^3+1\right)^2\mathrm{coth}\left(4x^3+1\right)\left(\frac{d}{dx}\left(4x^3\right)+0\right)$

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

$-2\mathrm{csch}\left(4x^3+1\right)^2\mathrm{coth}\left(4x^3+1\right)\frac{d}{dx}\left(4x^3\right)$
9

The derivative of the constant function ($1$) is equal to zero

$-2\mathrm{csch}\left(4x^3+1\right)^2\mathrm{coth}\left(4x^3+1\right)\frac{d}{dx}\left(4x^3\right)$

$-2\cdot 4\mathrm{csch}\left(4x^3+1\right)^2\mathrm{coth}\left(4x^3+1\right)\frac{d}{dx}\left(x^3\right)$

Multiply $-2$ times $4$

$-8\mathrm{csch}\left(4x^3+1\right)^2\mathrm{coth}\left(4x^3+1\right)\frac{d}{dx}\left(x^3\right)$
10

The derivative of a function multiplied by a constant ($4$) is equal to the constant times the derivative of the function

$-8\mathrm{csch}\left(4x^3+1\right)^2\mathrm{coth}\left(4x^3+1\right)\frac{d}{dx}\left(x^3\right)$

$-8\cdot 3x^{\left(3-1\right)}\mathrm{csch}\left(4x^3+1\right)^2\mathrm{coth}\left(4x^3+1\right)$

Subtract the values $3$ and $-1$

$-8\cdot 3x^{2}\mathrm{csch}\left(4x^3+1\right)^2\mathrm{coth}\left(4x^3+1\right)$

Multiply $-8$ times $3$

$-24x^{2}\mathrm{csch}\left(4x^3+1\right)^2\mathrm{coth}\left(4x^3+1\right)$
11

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

$-24x^{2}\mathrm{csch}\left(4x^3+1\right)^2\mathrm{coth}\left(4x^3+1\right)$

Final Answer

$-24x^{2}\mathrm{csch}\left(4x^3+1\right)^2\mathrm{coth}\left(4x^3+1\right)$

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