Math virtual assistant

Calculators Topics Go Premium About Snapxam
ENGESP

Inverse trigonometric functions differentiation Calculator

Get detailed solutions to your math problems with our Inverse trigonometric functions differentiation 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!

Go!
1
2
3
4
5
6
7
8
9
0
x
y
(◻)
◻/◻
÷
2

e
π
ln
log
log
lim
d/dx
Dx
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

1

Solved example of inverse trigonometric functions differentiation

$\frac{d}{dx}\left(\ln\left(\sqrt{\frac{1+\sin\left(x\right)}{1-\sin\left(x\right)}}\right)\right)$
2

The derivative of the natural logarithm of a function is equal to the derivative of the function divided by that function. If $f(x)=ln\:a$ (where $a$ is a function of $x$), then $\displaystyle f'(x)=\frac{a'}{a}$

$\frac{1}{\sqrt{\frac{1+\sin\left(x\right)}{1-\sin\left(x\right)}}}\cdot\frac{d}{dx}\left(\sqrt{\frac{1+\sin\left(x\right)}{1-\sin\left(x\right)}}\right)$
3

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

$\frac{1}{2}\left(\frac{1+\sin\left(x\right)}{1-\sin\left(x\right)}\right)^{-\frac{1}{2}}\left(\frac{1}{\sqrt{\frac{1+\sin\left(x\right)}{1-\sin\left(x\right)}}}\right)\frac{d}{dx}\left(\frac{1+\sin\left(x\right)}{1-\sin\left(x\right)}\right)$
4

Apply the formula: $a\frac{1}{x}$$=\frac{a}{x}$, where $a=\frac{1}{2}$ and $x=\sqrt{\frac{1+\sin\left(x\right)}{1-\sin\left(x\right)}}$

$\left(\frac{1+\sin\left(x\right)}{1-\sin\left(x\right)}\right)^{-\frac{1}{2}}\cdot\frac{\frac{1}{2}}{\sqrt{\frac{1+\sin\left(x\right)}{1-\sin\left(x\right)}}}\cdot\frac{d}{dx}\left(\frac{1+\sin\left(x\right)}{1-\sin\left(x\right)}\right)$
5

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 ${\displaystyle h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}}$

$\left(\frac{1+\sin\left(x\right)}{1-\sin\left(x\right)}\right)^{-\frac{1}{2}}\cdot\frac{\frac{1}{2}}{\sqrt{\frac{1+\sin\left(x\right)}{1-\sin\left(x\right)}}}\cdot\frac{\frac{d}{dx}\left(1+\sin\left(x\right)\right)\left(1-\sin\left(x\right)\right)-\left(1+\sin\left(x\right)\right)\frac{d}{dx}\left(1-\sin\left(x\right)\right)}{\left(1-\sin\left(x\right)\right)^2}$
6

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

$\left(\frac{1+\sin\left(x\right)}{1-\sin\left(x\right)}\right)^{-\frac{1}{2}}\cdot\frac{\frac{1}{2}}{\sqrt{\frac{1+\sin\left(x\right)}{1-\sin\left(x\right)}}}\cdot\frac{\left(\frac{d}{dx}\left(1\right)+\frac{d}{dx}\left(\sin\left(x\right)\right)\right)\left(1-\sin\left(x\right)\right)-\left(1+\sin\left(x\right)\right)\left(\frac{d}{dx}\left(1\right)+\frac{d}{dx}\left(-\sin\left(x\right)\right)\right)}{\left(1-\sin\left(x\right)\right)^2}$
7

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

$\left(\frac{1+\sin\left(x\right)}{1-\sin\left(x\right)}\right)^{-\frac{1}{2}}\cdot\frac{\frac{1}{2}}{\sqrt{\frac{1+\sin\left(x\right)}{1-\sin\left(x\right)}}}\cdot\frac{\left(0+\frac{d}{dx}\left(\sin\left(x\right)\right)\right)\left(1-\sin\left(x\right)\right)-\left(1+\sin\left(x\right)\right)\frac{d}{dx}\left(-\sin\left(x\right)\right)}{\left(1-\sin\left(x\right)\right)^2}$
8

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

$\left(\frac{1+\sin\left(x\right)}{1-\sin\left(x\right)}\right)^{-\frac{1}{2}}\cdot\frac{\frac{1}{2}}{\sqrt{\frac{1+\sin\left(x\right)}{1-\sin\left(x\right)}}}\cdot\frac{\frac{d}{dx}\left(\sin\left(x\right)\right)\left(1-\sin\left(x\right)\right)-\left(1+\sin\left(x\right)\right)\frac{d}{dx}\left(-\sin\left(x\right)\right)}{\left(1-\sin\left(x\right)\right)^2}$
9

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

$\left(\frac{1+\sin\left(x\right)}{1-\sin\left(x\right)}\right)^{-\frac{1}{2}}\cdot\frac{\frac{1}{2}}{\sqrt{\frac{1+\sin\left(x\right)}{1-\sin\left(x\right)}}}\cdot\frac{\frac{d}{dx}\left(\sin\left(x\right)\right)\left(1-\sin\left(x\right)\right)+\left(1+\sin\left(x\right)\right)\frac{d}{dx}\left(\sin\left(x\right)\right)}{\left(1-\sin\left(x\right)\right)^2}$
10

The derivative of the sine of a function is equal to the cosine of that function times the derivative of that function, in other words, if ${f(x) = \sin(x)}$, then ${f'(x) = \cos(x)\cdot D_x(x)}$

$\left(\frac{1+\sin\left(x\right)}{1-\sin\left(x\right)}\right)^{-\frac{1}{2}}\cdot\frac{\frac{1}{2}}{\sqrt{\frac{1+\sin\left(x\right)}{1-\sin\left(x\right)}}}\cdot\frac{\cos\left(x\right)\left(1-\sin\left(x\right)\right)+\left(1+\sin\left(x\right)\right)\cos\left(x\right)}{\left(1-\sin\left(x\right)\right)^2}$
11

The power of a quotient is equal to the quotient of the power of the numerator and denominator: $\displaystyle\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}$

$\left(\frac{1+\sin\left(x\right)}{1-\sin\left(x\right)}\right)^{-\frac{1}{2}}\cdot\frac{\frac{1}{2}}{\frac{\sqrt{1+\sin\left(x\right)}}{\sqrt{1-\sin\left(x\right)}}}\cdot\frac{\cos\left(x\right)\left(1-\sin\left(x\right)\right)+\left(1+\sin\left(x\right)\right)\cos\left(x\right)}{\left(1-\sin\left(x\right)\right)^2}$
12

Simplifying the fraction

$\frac{1}{2}\left(\frac{1+\sin\left(x\right)}{1-\sin\left(x\right)}\right)^{-\frac{1}{2}}\left(\frac{\sqrt{1-\sin\left(x\right)}}{\sqrt{1+\sin\left(x\right)}}\right)\left(\frac{\cos\left(x\right)\left(1-\sin\left(x\right)\right)+\left(1+\sin\left(x\right)\right)\cos\left(x\right)}{\left(1-\sin\left(x\right)\right)^2}\right)$
13

The power of a quotient is equal to the quotient of the power of the numerator and denominator: $\displaystyle\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}$

$\frac{1}{2}\left(\frac{\sqrt{1-\sin\left(x\right)}}{\sqrt{1+\sin\left(x\right)}}\right)\left(\frac{\left(1+\sin\left(x\right)\right)^{-\frac{1}{2}}}{\left(1-\sin\left(x\right)\right)^{-\frac{1}{2}}}\right)\left(\frac{\cos\left(x\right)\left(1-\sin\left(x\right)\right)+\left(1+\sin\left(x\right)\right)\cos\left(x\right)}{\left(1-\sin\left(x\right)\right)^2}\right)$
14

Multiplying the fraction and term

$\frac{\frac{1}{2}\sqrt{1-\sin\left(x\right)}}{\sqrt{1+\sin\left(x\right)}}\cdot\frac{\left(1+\sin\left(x\right)\right)^{-\frac{1}{2}}}{\left(1-\sin\left(x\right)\right)^{-\frac{1}{2}}}\cdot\frac{\cos\left(x\right)\left(1-\sin\left(x\right)\right)+\left(1+\sin\left(x\right)\right)\cos\left(x\right)}{\left(1-\sin\left(x\right)\right)^2}$

Struggling with math?

Access detailed step by step solutions to millions of problems, growing every day!