In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.
12. y = ln(10/x)

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Recognize that the function is given as \(y = \ln\left(\frac{10}{x}\right)\), which is a natural logarithm of a quotient.

Use the logarithm property that \(\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)\) to rewrite the function as \(y = \ln(10) - \ln(x)\).

Note that \(\ln(10)\) is a constant since 10 is a constant, so its derivative with respect to \(x\) is zero.

Recall the derivative of \(\ln(x)\) with respect to \(x\) is \(\frac{1}{x}\), so the derivative of \(-\ln(x)\) is \(-\frac{1}{x}\).

Combine these results to write the derivative \(\frac{dy}{dx} = 0 - \frac{1}{x} = -\frac{1}{x}\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Derivative of the Natural Logarithm Function

The derivative of ln(u), where u is a differentiable function of x, is (1/u) times the derivative of u. This rule allows us to differentiate logarithmic functions by applying the chain rule to the inner function.

Chain Rule

The chain rule is used to differentiate composite functions. It states that the derivative of a function composed of another function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.

Derivative of a Quotient

When differentiating a function expressed as a quotient, such as 10/x, it can be rewritten as 10x⁻¹. Differentiating powers of x involves applying the power rule, which states that d/dx of x^n is n*x^(n-1).

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