15–48. Derivatives Find the derivative of the following functions.
y = 10^x(In 10^x-1)

Verified step by step guidance

1. Identify the function for which you need to find the derivative: \( y = 10^x (\ln(10^x - 1)) \). This is a product of two functions, so you will use the product rule.
2. Recall the product rule for derivatives: If \( u(x) \) and \( v(x) \) are functions of \( x \), then the derivative of their product \( u(x)v(x) \) is \( u'(x)v(x) + u(x)v'(x) \).
3. Assign \( u(x) = 10^x \) and \( v(x) = \ln(10^x - 1) \). Find the derivative of \( u(x) \), which is \( u'(x) = 10^x \ln(10) \) because the derivative of \( 10^x \) is \( 10^x \ln(10) \).
4. Find the derivative of \( v(x) = \ln(10^x - 1) \). Use the chain rule: the derivative of \( \ln(g(x)) \) is \( \frac{1}{g(x)} \cdot g'(x) \). Here, \( g(x) = 10^x - 1 \), so \( g'(x) = 10^x \ln(10) \). Thus, \( v'(x) = \frac{10^x \ln(10)}{10^x - 1} \).
5. Apply the product rule: \( y' = u'(x)v(x) + u(x)v'(x) = 10^x \ln(10) \cdot \ln(10^x - 1) + 10^x \cdot \frac{10^x \ln(10)}{10^x - 1} \). Simplify this expression to find the derivative.

Key Concepts

Derivatives

A derivative represents the rate of change of a function with respect to its variable. It is a fundamental concept in calculus that allows us to determine how a function behaves at any given point. The derivative can be interpreted as the slope of the tangent line to the curve of the function at a specific point.

Exponential Functions

Exponential functions are mathematical functions of the form y = a^x, where 'a' is a constant and 'x' is the variable. In the context of the given function, 10^x is an exponential function, and its derivative can be found using the rule that states the derivative of a^x is a^x * ln(a). Understanding how to differentiate exponential functions is crucial for solving the problem.

Product Rule

The product rule is a formula used to find the derivative of the product of two functions. It states that if u(x) and v(x) are two differentiable functions, then the derivative of their product is given by u'v + uv'. In the given function, the presence of the product of 10^x and (ln(10^x) - 1) necessitates the use of the product rule to find the derivative correctly.