To differentiate the function [tex]\( y = \ln \left(x^5 \sin^4 x\right) \)[/tex], we will use the properties of logarithms and the rules of differentiation. Here are the steps:
1. Simplify the Logarithmic Expression:
We use the properties of logarithms to simplify the function:
[tex]\[
y = \ln \left(x^5 \sin^4 x\right)
\][/tex]
Applying the logarithmic property [tex]\(\ln (a \cdot b) = \ln a + \ln b\)[/tex], we get:
[tex]\[
y = \ln (x^5) + \ln (\sin^4 x)
\][/tex]
Further simplifying using [tex]\(\ln (a^b) = b \ln a\)[/tex], we get:
[tex]\[
y = 5 \ln (x) + 4 \ln (\sin x)
\][/tex]
2. Apply the Differentiation Rules:
Now, we need to differentiate each term with respect to [tex]\( x \)[/tex].
- The derivative of [tex]\( 5 \ln (x) \)[/tex]:
[tex]\[
\frac{d}{dx} [5 \ln (x)] = 5 \cdot \frac{1}{x} = \frac{5}{x}
\][/tex]
- The derivative of [tex]\( 4 \ln (\sin x) \)[/tex]:
[tex]\[
\frac{d}{dx} [4 \ln (\sin x)] = 4 \cdot \frac{1}{\sin x} \cdot \frac{d}{dx} [\sin x] = 4 \cdot \frac{1}{\sin x} \cdot \cos x = 4 \cdot \cot x
\][/tex]
3. Combine the Results:
Adding the derivatives of the two terms, we get:
[tex]\[
\frac{dy}{dx} = \frac{5}{x} + 4 \cot x
\][/tex]
Combining these terms into a single fraction:
[tex]\[
\frac{dy}{dx} = \frac{4 x^5 \sin^3 x \cos x + 5 x^4 \sin^4 x}{x^5 \sin^4 x}
\][/tex]
Therefore, the derivative of [tex]\( y = \ln \left(x^5 \sin^4 x\right) \)[/tex] is:
[tex]\[
\frac{dy}{dx} = \frac{4 x^5 \sin^3 x \cos x + 5 x^4 \sin^4 x}{x^5 \sin^4 x}
\][/tex]
