In Exercises 59–86, find the derivative of y with respect to the given independent variable.
75. y = x³ log₁₀ x

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Identify the function given: \(y = x^{3} \log_{10} x\). Notice that this is a product of two functions of \(x\): \(x^{3}\) and \(\log_{10} x\).

Recall that to differentiate a product of two functions, we use the product rule: \(\frac{d}{dx}[u \cdot v] = u'v + uv'\), where \(u = x^{3}\) and \(v = \log_{10} x\).

Find the derivative of \(u = x^{3}\). Using the power rule, \(u' = 3x^{2}\).

Find the derivative of \(v = \log_{10} x\). Remember that \(\log_{10} x = \frac{\ln x}{\ln 10}\), so its derivative is \(v' = \frac{1}{x \ln 10}\).

Apply the product rule: \(\frac{dy}{dx} = u'v + uv' = 3x^{2} \log_{10} x + x^{3} \cdot \frac{1}{x \ln 10}\). Simplify the expression where possible.

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Key Concepts

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

Derivative of a Power Function

The derivative of a power function x^n is found using the power rule, which states that d/dx (x^n) = n*x^(n-1). For example, the derivative of x³ is 3x². This rule is fundamental when differentiating terms involving powers of x.

Derivative of Logarithmic Functions

The derivative of the logarithm logₐ(x) with base a is 1/(x ln(a)), where ln(a) is the natural logarithm of the base. For log₁₀(x), the derivative is 1/(x ln(10)). Understanding this helps differentiate terms involving logarithms with bases other than e.

Product Rule for Differentiation

When differentiating a product of two functions, f(x) and g(x), the product rule states that d/dx [f(x)g(x)] = f'(x)g(x) + f(x)g'(x). This rule is essential for differentiating expressions like y = x³ log₁₀(x), where both factors depend on x.

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