Ch. 7 - Transcendental Functions

Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo

Hass 15th Edition

Ch. 7 - Transcendental Functions

Problema 7.2.8

Capitolo 7, Problema 7.2.8

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

Guida verificata passo dopo passo

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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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.

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Derivative of the Natural Logarithmic Function

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.

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The Power Rule

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.

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Intro to the Chain Rule

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