Find the derivative of arccos(sin(x))

\frac{d}{dx}\left(arccos\left(\sin\left(x\right)\right)\right)

Go!
1
2
3
4
5
6
7
8
9
0
x
y
(◻)
◻/◻
2

e
π
ln
log
lim
d/dx
d/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

Answer

$-1$

Step by step solution

Problem

$\frac{d}{dx}\left(arccos\left(\sin\left(x\right)\right)\right)$
1

Taking the derivative of arccosine

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

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 ${f'(x) = \cos(x)\cdot D_x(x)}$

$\cos\left(x\right)\frac{-1}{\sqrt{1-\sin\left(x\right)^2}}$
3

Applying the trigonometric identity: $1-\sin\left(\theta\right)^2=\cos\left(\theta\right)^2$

$\cos\left(x\right)\frac{-1}{\sqrt{\cos\left(x\right)^2}}$
4

Applying the power of a power property

$\cos\left(x\right)\frac{-1}{\cos\left(x\right)}$
5

Multiplying the fraction and term

$\frac{-\cos\left(x\right)}{\cos\left(x\right)}$
6

Simplifying the fraction by $\cos\left(x\right)$

$-1$

Answer

$-1$

Problem Analysis

Main topic:

Differential calculus

Time to solve it:

0.25 seconds

Views:

109