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# Derivatives of hyperbolic trigonometric functions Calculator

## Get detailed solutions to your math problems with our Derivatives of hyperbolic trigonometric functions step-by-step calculator. Practice your math skills and learn step by step with our math solver. Check out all of our online calculators here.

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###  Difficult Problems

1

Here, we show you a step-by-step solved example of derivatives of hyperbolic trigonometric functions. This solution was automatically generated by our smart calculator:

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

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

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

Add the values $2$ and $-1$

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

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

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

Subtract the values $2$ and $-1$

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

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

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

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

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

Taking the derivative of hyperbolic cosecant

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

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

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

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

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

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

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

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

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

Multiply $-2$ times $4$

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

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

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

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

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

Subtract the values $3$ and $-1$

$-24\mathrm{csch}\left(4x^3+1\right)^2x^{2}\mathrm{coth}\left(4x^3+1\right)$
9

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

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

Multiply $-8$ times $3$

$-24\mathrm{csch}\left(4x^3+1\right)^2x^{2}\mathrm{coth}\left(4x^3+1\right)$
10

Expand and simplify

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

##  Final answer to the problem

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

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