Questions: Use logarithmic differentiation to find the derivative of y with respect to the given independent variable. y=(sin 5 x)^(7 x) dy/dx=

Use logarithmic differentiation to find the derivative of y with respect to the given independent variable. y=(sin 5 x)^(7 x) dy/dx=
Transcript text: Use logarithmic differentiation to find the derivative of $y$ with respect to the given independent variable. \[ \begin{array}{r} y=(\sin 5 x)^{7 x} \\ \frac{d y}{d x}=\square \end{array} \] $\square$
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Solution

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Solution Steps

To find the derivative of \( y = (\sin 5x)^{7x} \) using logarithmic differentiation, follow these steps:

  1. Take the natural logarithm of both sides to simplify the expression.
  2. Differentiate both sides with respect to \( x \).
  3. Solve for \( \frac{dy}{dx} \).
Step 1: Define the Function

We start with the function given by \[ y = (\sin(5x))^{7x}. \]

Step 2: Take the Natural Logarithm

Taking the natural logarithm of both sides, we have: \[ \ln(y) = 7x \ln(\sin(5x)). \]

Step 3: Differentiate Both Sides

Differentiating both sides with respect to \( x \): \[ \frac{d}{dx}(\ln(y)) = \frac{d}{dx}(7x \ln(\sin(5x))). \] Using the product rule and chain rule, we find: \[ \frac{1}{y} \frac{dy}{dx} = 7 \ln(\sin(5x)) + 7x \cdot \frac{5 \cos(5x)}{\sin(5x)}. \]

Step 4: Solve for \(\frac{dy}{dx}\)

Multiplying both sides by \( y \) gives: \[ \frac{dy}{dx} = y \left( 7 \ln(\sin(5x)) + 35x \frac{\cos(5x)}{\sin(5x)} \right). \] Substituting back \( y = (\sin(5x))^{7x} \): \[ \frac{dy}{dx} = (\sin(5x))^{7x} \left( 7 \ln(\sin(5x)) + 35x \frac{\cos(5x)}{\sin(5x)} \right). \]

Step 5: Simplify the Expression

The expression can be simplified to: \[ \frac{dy}{dx} = 7 \left( 5x \cos(5x) + \ln(\sin(5x)) \sin(5x) \right) (\sin(5x))^{7x - 1}. \]

Final Answer

Thus, the derivative of \( y \) with respect to \( x \) is: \[ \boxed{\frac{dy}{dx} = 7 \left( 5x \cos(5x) + \ln(\sin(5x)) \sin(5x) \right) (\sin(5x))^{7x - 1}}. \]

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