Answer :
To find the derivative of [tex]\( y = x^4 \sin(x) \)[/tex] using logarithmic differentiation, let's follow these steps carefully.
### Step 1: Take the natural logarithm of both sides
Start by taking the natural logarithm of the given function:
[tex]\[ \ln(y) = \ln(x^4 \sin(x)) \][/tex]
### Step 2: Use properties of logarithms to simplify
Utilize the property of logarithms that [tex]\(\ln(a \cdot b) = \ln(a) + \ln(b)\)[/tex]:
[tex]\[ \ln(y) = \ln(x^4) + \ln(\sin(x)) \][/tex]
Further simplify using the power rule of logarithms, where [tex]\(\ln(a^b) = b \cdot \ln(a)\)[/tex]:
[tex]\[ \ln(y) = 4 \ln(x) + \ln(\sin(x)) \][/tex]
### Step 3: Differentiate both sides with respect to x
Differentiate both sides of the equation with respect to [tex]\( x \)[/tex]. Remember to use the chain rule on the left-hand side and the product rule wherever necessary on the right-hand side.
#### Differentiating [tex]\(\ln(y)\)[/tex]:
[tex]\[ \frac{d}{dx}[\ln(y)] = \frac{1}{y} \cdot \frac{dy}{dx} \][/tex]
#### Differentiating [tex]\(4 \ln(x)\)[/tex]:
[tex]\[ \frac{d}{dx}[4 \ln(x)] = 4 \cdot \frac{1}{x} = \frac{4}{x} \][/tex]
#### Differentiating [tex]\(\ln(\sin(x))\)[/tex]:
[tex]\[ \frac{d}{dx}[\ln(\sin(x))] = \frac{1}{\sin(x)} \cdot \cos(x) = \cot(x) \][/tex]
Combining these results, we have:
[tex]\[ \frac{1}{y} \cdot \frac{dy}{dx} = \frac{4}{x} + \cot(x) \][/tex]
### Step 4: Solve for [tex]\( \frac{dy}{dx} \)[/tex]
Multiply both sides of the equation by [tex]\( y \)[/tex] to solve for [tex]\( \frac{dy}{dx} \)[/tex]:
[tex]\[ \frac{dy}{dx} = y \left( \frac{4}{x} + \cot(x) \right) \][/tex]
### Step 5: Substitute [tex]\( y = x^4 \sin(x) \)[/tex]
Replace [tex]\( y \)[/tex] with the original function [tex]\( x^4 \sin(x) \)[/tex]:
[tex]\[ \frac{dy}{dx} = x^4 \sin(x) \left( \frac{4}{x} + \cot(x) \right) \][/tex]
### Step 6: Simplify the expression
Expand and simplify the final expression:
[tex]\[ \frac{dy}{dx} = x^4 \sin(x) \left( \frac{4}{x} + \frac{\cos(x)}{\sin(x)} \right) \][/tex]
[tex]\[ \frac{dy}{dx} = x^4 \left( \frac{4 \sin(x)}{x} + \cos(x) \right) \][/tex]
[tex]\[ \frac{dy}{dx} = x^4 \left( \frac{4 \sin(x)}{x} + \cos(x) \right) \][/tex]
[tex]\[ \frac{dy}{dx} = 4x^3 \sin(x) + x^4 \cos(x) \][/tex]
Hence, the derivative of the function [tex]\( y = x^4 \sin(x) \)[/tex] is:
[tex]\[ y'(x) = x^4 \cos(x) + 4 x^3 \sin(x) \][/tex]
### Step 1: Take the natural logarithm of both sides
Start by taking the natural logarithm of the given function:
[tex]\[ \ln(y) = \ln(x^4 \sin(x)) \][/tex]
### Step 2: Use properties of logarithms to simplify
Utilize the property of logarithms that [tex]\(\ln(a \cdot b) = \ln(a) + \ln(b)\)[/tex]:
[tex]\[ \ln(y) = \ln(x^4) + \ln(\sin(x)) \][/tex]
Further simplify using the power rule of logarithms, where [tex]\(\ln(a^b) = b \cdot \ln(a)\)[/tex]:
[tex]\[ \ln(y) = 4 \ln(x) + \ln(\sin(x)) \][/tex]
### Step 3: Differentiate both sides with respect to x
Differentiate both sides of the equation with respect to [tex]\( x \)[/tex]. Remember to use the chain rule on the left-hand side and the product rule wherever necessary on the right-hand side.
#### Differentiating [tex]\(\ln(y)\)[/tex]:
[tex]\[ \frac{d}{dx}[\ln(y)] = \frac{1}{y} \cdot \frac{dy}{dx} \][/tex]
#### Differentiating [tex]\(4 \ln(x)\)[/tex]:
[tex]\[ \frac{d}{dx}[4 \ln(x)] = 4 \cdot \frac{1}{x} = \frac{4}{x} \][/tex]
#### Differentiating [tex]\(\ln(\sin(x))\)[/tex]:
[tex]\[ \frac{d}{dx}[\ln(\sin(x))] = \frac{1}{\sin(x)} \cdot \cos(x) = \cot(x) \][/tex]
Combining these results, we have:
[tex]\[ \frac{1}{y} \cdot \frac{dy}{dx} = \frac{4}{x} + \cot(x) \][/tex]
### Step 4: Solve for [tex]\( \frac{dy}{dx} \)[/tex]
Multiply both sides of the equation by [tex]\( y \)[/tex] to solve for [tex]\( \frac{dy}{dx} \)[/tex]:
[tex]\[ \frac{dy}{dx} = y \left( \frac{4}{x} + \cot(x) \right) \][/tex]
### Step 5: Substitute [tex]\( y = x^4 \sin(x) \)[/tex]
Replace [tex]\( y \)[/tex] with the original function [tex]\( x^4 \sin(x) \)[/tex]:
[tex]\[ \frac{dy}{dx} = x^4 \sin(x) \left( \frac{4}{x} + \cot(x) \right) \][/tex]
### Step 6: Simplify the expression
Expand and simplify the final expression:
[tex]\[ \frac{dy}{dx} = x^4 \sin(x) \left( \frac{4}{x} + \frac{\cos(x)}{\sin(x)} \right) \][/tex]
[tex]\[ \frac{dy}{dx} = x^4 \left( \frac{4 \sin(x)}{x} + \cos(x) \right) \][/tex]
[tex]\[ \frac{dy}{dx} = x^4 \left( \frac{4 \sin(x)}{x} + \cos(x) \right) \][/tex]
[tex]\[ \frac{dy}{dx} = 4x^3 \sin(x) + x^4 \cos(x) \][/tex]
Hence, the derivative of the function [tex]\( y = x^4 \sin(x) \)[/tex] is:
[tex]\[ y'(x) = x^4 \cos(x) + 4 x^3 \sin(x) \][/tex]
