ðŸ‘‰ Try now NerdPal! Our new math app on iOS and Android

## Get detailed solutions to your math problems with our Advanced 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!
Symbolic mode
Text mode
Go!
1
2
3
4
5
6
7
8
9
0
a
b
c
d
f
g
m
n
u
v
w
x
y
z
.
(◻)
+
-
×
◻/◻
/
÷
2

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

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

###  Difficult Problems

1

Here, we show you a step-by-step solved example of advanced differentiation. This solution was automatically generated by our smart calculator:

$\frac{d}{dx}\left(sin\left(x\right)^{ln\left(x\right)}\right)$
2

To derive the function $\sin\left(x\right)^{\ln\left(x\right)}$, use the method of logarithmic differentiation. First, assign the function to $y$, then take the natural logarithm of both sides of the equation

$y=\sin\left(x\right)^{\ln\left(x\right)}$
3

Apply natural logarithm to both sides of the equality

$\ln\left(y\right)=\ln\left(\sin\left(x\right)^{\ln\left(x\right)}\right)$
4

Using the power rule of logarithms: $\log_a(x^n)=n\cdot\log_a(x)$

$\ln\left(y\right)=\ln\left(x\right)\ln\left(\sin\left(x\right)\right)$
5

Derive both sides of the equality with respect to $x$

$\frac{d}{dx}\left(\ln\left(y\right)\right)=\frac{d}{dx}\left(\ln\left(x\right)\ln\left(\sin\left(x\right)\right)\right)$
6

Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=$\frac{d}{dx}\left(\ln\left(y\right)\right)=\frac{d}{dx}\left(\ln\left(x\right)\right)\ln\left(\sin\left(x\right)\right)+\ln\left(x\right)\frac{d}{dx}\left(\ln\left(\sin\left(x\right)\right)\right)$The derivative of the natural logarithm of a function is equal to the derivative of the function divided by that function. If$f(x)=ln\:a$(where$a$is a function of$x$), then$\displaystyle f'(x)=\frac{a'}{a}\frac{1}{y}\frac{d}{dx}\left(y\right)=\frac{1}{x}\frac{d}{dx}\left(x\right)\ln\left(\sin\left(x\right)\right)+\ln\left(x\right)\frac{d}{dx}\left(\ln\left(\sin\left(x\right)\right)\right)$The derivative of the natural logarithm of a function is equal to the derivative of the function divided by that function. If$f(x)=ln\:a$(where$a$is a function of$x$), then$\displaystyle f'(x)=\frac{a'}{a}\frac{1}{y}\frac{d}{dx}\left(y\right)=\frac{1}{x}\frac{d}{dx}\left(x\right)\ln\left(\sin\left(x\right)\right)+\ln\left(x\right)\frac{1}{\sin\left(x\right)}\frac{d}{dx}\left(\sin\left(x\right)\right)$7 The derivative of the natural logarithm of a function is equal to the derivative of the function divided by that function. If$f(x)=ln\:a$(where$a$is a function of$x$), then$\displaystyle f'(x)=\frac{a'}{a}\frac{1}{y}\frac{d}{dx}\left(y\right)=\frac{1}{x}\frac{d}{dx}\left(x\right)\ln\left(\sin\left(x\right)\right)+\ln\left(x\right)\frac{1}{\sin\left(x\right)}\frac{d}{dx}\left(\sin\left(x\right)\right)$The derivative of the linear function is equal to$11\left(\frac{1}{y}\right)$Any expression multiplied by$1$is equal to itself$\frac{1}{y}$8 The derivative of the linear function is equal to$1\frac{y^{\prime}}{y}=\frac{1}{x}\frac{d}{dx}\left(x\right)\ln\left(\sin\left(x\right)\right)+\ln\left(x\right)\frac{1}{\sin\left(x\right)}\frac{d}{dx}\left(\sin\left(x\right)\right)$The derivative of the linear function is equal to$11\left(\frac{1}{y}\right)$Any expression multiplied by$1$is equal to itself$\frac{1}{y}$The derivative of the linear function is equal to$11\left(\frac{1}{x}\right)\ln\left(\sin\left(x\right)\right)$Any expression multiplied by$1$is equal to itself$\frac{1}{x}\ln\left(\sin\left(x\right)\right)$9 The derivative of the linear function is equal to$1\frac{y^{\prime}}{y}=\frac{1}{x}\ln\left(\sin\left(x\right)\right)+\ln\left(x\right)\frac{1}{\sin\left(x\right)}\frac{d}{dx}\left(\sin\left(x\right)\right)$10 Multiply the fraction and term$\frac{y^{\prime}}{y}=\frac{\ln\left(\sin\left(x\right)\right)}{x}+\ln\left(x\right)\frac{1}{\sin\left(x\right)}\frac{d}{dx}\left(\sin\left(x\right)\right)$11 The derivative of the sine of a function is equal to the cosine of that function times the derivative of that function, in other words, if${f(x) = \sin(x)}$, then${f'(x) = \cos(x)\cdot D_x(x)}\frac{y^{\prime}}{y}=\frac{\ln\left(\sin\left(x\right)\right)}{x}+\frac{d}{dx}\left(x\right)\ln\left(x\right)\frac{1}{\sin\left(x\right)}\cos\left(x\right)$The derivative of the linear function is equal to$11\left(\frac{1}{y}\right)$Any expression multiplied by$1$is equal to itself$\frac{1}{y}$The derivative of the linear function is equal to$11\left(\frac{1}{x}\right)\ln\left(\sin\left(x\right)\right)$Any expression multiplied by$1$is equal to itself$\frac{1}{x}\ln\left(\sin\left(x\right)\right)$The derivative of the linear function is equal to$11\ln\left(x\right)\frac{1}{\sin\left(x\right)}\cos\left(x\right)$Any expression multiplied by$1$is equal to itself$\ln\left(x\right)\frac{1}{\sin\left(x\right)}\cos\left(x\right)$12 The derivative of the linear function is equal to$1\frac{y^{\prime}}{y}=\frac{\ln\left(\sin\left(x\right)\right)}{x}+\ln\left(x\right)\frac{1}{\sin\left(x\right)}\cos\left(x\right)$13 Multiply the fraction and term$\frac{y^{\prime}}{y}=\frac{\ln\left(\sin\left(x\right)\right)}{x}+\frac{\ln\left(x\right)\cos\left(x\right)}{\sin\left(x\right)}$14 Multiply both sides of the equation by$yy^{\prime}=y\left(\frac{\ln\left(\sin\left(x\right)\right)}{x}+\frac{\ln\left(x\right)\cos\left(x\right)}{\sin\left(x\right)}\right)$15 Substitute$y$for the original function:$\sin\left(x\right)^{\ln\left(x\right)}y^{\prime}=\left(\frac{\ln\left(\sin\left(x\right)\right)}{x}+\frac{\ln\left(x\right)\cos\left(x\right)}{\sin\left(x\right)}\right)\sin\left(x\right)^{\ln\left(x\right)}$16 The derivative of the function results in$\left(\frac{\ln\left(\sin\left(x\right)\right)}{x}+\frac{\ln\left(x\right)\cos\left(x\right)}{\sin\left(x\right)}\right)\sin\left(x\right)^{\ln\left(x\right)}$Apply the trigonometric identity:$\frac{\cos\left(\theta \right)}{\sin\left(\theta \right)}=\cot\left(\theta \right)\left(\frac{\ln\left(\sin\left(x\right)\right)}{x}+\ln\left(x\right)\cot\left(x\right)\right)\sin\left(x\right)^{\ln\left(x\right)}$17 Simplify the derivative$\left(\frac{\ln\left(\sin\left(x\right)\right)}{x}+\ln\left(x\right)\cot\left(x\right)\right)\sin\left(x\right)^{\ln\left(x\right)}$##  Final answer to the problem$\left(\frac{\ln\left(\sin\left(x\right)\right)}{x}+\ln\left(x\right)\cot\left(x\right)\right)\sin\left(x\right)^{\ln\left(x\right)}\$

### Are you struggling with math?

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