In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.
8. y = ln kx, k constant

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Identify the function given: $y = \ln(kx)$, where $k$ is a constant and $x$ is the variable with respect to which we differentiate.

Recall the derivative rule for the natural logarithm function: if $y = \ln(u)$, then $\frac{dy}{dx} = \frac{1}{u} \cdot \frac{du}{dx}$.

Set $u = kx$. Since $k$ is a constant, find the derivative of $u$ with respect to $x$: $\frac{du}{dx} = k$.

Apply the chain rule: $\frac{dy}{dx} = \frac{1}{kx} \cdot k$.

Simplify the expression to write the derivative in its simplest form.

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주요 개념

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Derivative of the Natural Logarithm Function

The derivative of the natural logarithm function ln(u) with respect to its variable is 1/u times the derivative of u. This rule is essential for differentiating expressions involving ln, such as ln(kx), where the chain rule applies.

Constant Multiple Rule

When differentiating a function multiplied by a constant, the constant can be factored out and remains unchanged. For example, in ln(kx), k is a constant multiplier inside the logarithm, affecting the differentiation through the chain rule but not changing the derivative directly.

Chain Rule

The chain rule is used to differentiate composite functions. For y = ln(kx), the outer function is ln(u) and the inner function is u = kx. The derivative is found by multiplying the derivative of the outer function evaluated at the inner function by the derivative of the inner function.

In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.8. y = ln kx, k constant