find dy/dx if y=ln5x

find dy/dx if y=lnx/5

find dy/dx if y=sin^-1(x³)

To find the derivative \( \frac{dy}{dx} \) for the given functions, we will apply the rules of differentiation, particularly focusing on the natural logarithm and inverse sine functions. Let’s break down each function step by step.

1. Derivative of \( y = \ln(5x) \)

For the function \( y = \ln(5x) \), we can use the chain rule and the property of logarithms. The property states that \( \ln(ab) = \ln(a) + \ln(b) \). Thus, we can rewrite the function:

• Using the property: \( y = \ln(5) + \ln(x) \)

Now, we differentiate:

• Since \( \ln(5) \) is a constant, its derivative is 0.
• The derivative of \( \ln(x) \) is \( \frac{1}{x} \).

Putting it all together, we have:

Result: \( \frac{dy}{dx} = \frac{1}{x} \)

2. Derivative of \( y = \frac{\ln(x)}{5} \)

For the function \( y = \frac{\ln(x)}{5} \), we can factor out the constant \( \frac{1}{5} \) when differentiating:

• So, \( y = \frac{1}{5} \ln(x) \)

Now, we differentiate:

• The derivative of \( \ln(x) \) is \( \frac{1}{x} \).

Thus, applying the constant factor rule:

Result: \( \frac{dy}{dx} = \frac{1}{5} \cdot \frac{1}{x} = \frac{1}{5x} \)

3. Derivative of \( y = \sin^{-1}(x^3) \)

For the function \( y = \sin^{-1}(x^3) \), we will again use the chain rule. The derivative of \( \sin^{-1}(u) \) is \( \frac{1}{\sqrt{1-u^2}} \), where \( u = x^3 \).

• First, we differentiate the outer function: \( \frac{1}{\sqrt{1-(x^3)^2}} = \frac{1}{\sqrt{1-x^6}} \)
• Next, we differentiate the inner function \( u = x^3 \), which gives us \( \frac{du}{dx} = 3x^2 \).

Now, applying the chain rule:

Result: \( \frac{dy}{dx} = \frac{1}{\sqrt{1-x^6}} \cdot 3x^2 = \frac{3x^2}{\sqrt{1-x^6}} \)

Summary of Results

• For \( y = \ln(5x) \): \( \frac{dy}{dx} = \frac{1}{x} \)
• For \( y = \frac{\ln(x)}{5} \): \( \frac{dy}{dx} = \frac{1}{5x} \)
• For \( y = \sin^{-1}(x^3) \): \( \frac{dy}{dx} = \frac{3x^2}{\sqrt{1-x^6}} \)

These derivatives provide insight into the rate of change of each function with respect to \( x \). If you have any further questions or need clarification on any of the steps, feel free to ask!