# Step-by-step Solution

## Find the derivative using the product rule (d/dx)(ln(sin(x))sin(cos(x)))

Go
1
2
3
4
5
6
7
8
9
0
x
y
(◻)
◻/◻
2

e
π
ln
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

### Videos

$\cos\left(x\right)\sin\left(\cos\left(x\right)\right)\csc\left(x\right)-\ln\left(\sin\left(x\right)\right)\cos\left(\cos\left(x\right)\right)\sin\left(x\right)$

## Step-by-step explanation

Problem to solve:

$\frac{d}{dx}\left(\ln\left(\sin\left(x\right)\right)\cdot\sin\left(\cos\left(x\right)\right)\right)$
1

Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=\ln\left(\sin\left(x\right)\right)$ and $g=\sin\left(\cos\left(x\right)\right)$

$\frac{d}{dx}\left(\ln\left(\sin\left(x\right)\right)\right)\sin\left(\cos\left(x\right)\right)+\ln\left(\sin\left(x\right)\right)\frac{d}{dx}\left(\sin\left(\cos\left(x\right)\right)\right)$
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 ${f'(x) = \cos(x)\cdot D_x(x)}$

$\frac{d}{dx}\left(\ln\left(\sin\left(x\right)\right)\right)\sin\left(\cos\left(x\right)\right)+\ln\left(\sin\left(x\right)\right)\cos\left(\cos\left(x\right)\right)\frac{d}{dx}\left(\cos\left(x\right)\right)$

$\cos\left(x\right)\sin\left(\cos\left(x\right)\right)\csc\left(x\right)-\ln\left(\sin\left(x\right)\right)\cos\left(\cos\left(x\right)\right)\sin\left(x\right)$
$\frac{d}{dx}\left(\ln\left(\sin\left(x\right)\right)\cdot\sin\left(\cos\left(x\right)\right)\right)$

### Main topic:

Differential calculus

~ 1.38 seconds

### Struggling with math?

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