Problem 4 - Technical Mathematics with Calculus

Question

Derivative of \(b^{u}\). Differentiate. $$y=10^{3 x}$$

Step-by-Step Solution

Answer

The derivative of the function y = 10^(3x) is y' = 3 * ln(10) * 10^(3x).

Step 1: Identify the function to differentiate

Recognize that the function to differentiate is of the form where a constant base is raised to a variable power. Specifically, the function we're differentiating is y = 10^(3x). Here, the base is 10, and the exponent is 3x.

Step 2: Apply the chain rule for differentiation

The chain rule states that if you have a composite function f(g(x)), its derivative is f'(g(x))g'(x). In this case, our f is 10^u and our g(x) is 3x. Therefore, f'(u) = ln(10) * 10^u, and g'(x) = 3.

Step 3: Chain the derivatives together

Now we chain the derivatives together as per the chain rule: the derivative of y with respect to x is f'(g(x))*g'(x). This gives us the derivative y' = ln(10) * 10^(3x) * 3.

Step 4: Simplify the expression

We find that the derivative, in its simplest form, is y' = 3 * ln(10) * 10^(3x).

Key Concepts

Chain Rule for Differentiation

The chain rule for differentiation is a fundamental tool in calculus for finding the derivative of composite functions. A composite function is formed when one function is applied to the result of another function. Suppose we have two functions, f(u) and g(x), such that our composite function is f(g(x)). The chain rule tells us that the derivative of this composite function with respect to x is given by multiplying the derivative of f with respect to u (f'(u)) by the derivative of g with respect to x (g'(x)). In formula terms, this can be expressed as:
\[ \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x) \]
Understanding and applying the chain rule is crucial for dealing with more complex functions involving exponents, products, or quotients where one part of the function affects another.

Derivative of Exponential Functions

Exponential functions have the form f(x) = a^x, where 'a' is a constant and 'x' is an exponent variable. When differentiating exponential functions, the base 'a' of the exponent plays a significant role. If the base is e (Euler's number, approximately equal to 2.718281828), the function simply differentiates to itself, f'(x) = e^x. However, when the base is not e, we can use a formula that involves the natural logarithm of the base. The derivative of a^x is given by:
\[ f'(x) = \ln(a) \cdot a^x \]
This formula is derived from the fact that any exponential function with base a can be rewritten in terms of base e, using the property that a^x = (e^\ln(a))^x.

Natural Logarithm Properties

The natural logarithm, usually written as \(\ln(x)\), has properties that are particularly useful when dealing with exponential functions. Some key properties of natural logarithms include:

• The logarithm of a product is the sum of the logarithms: \(\ln(ab) = \ln(a) + \ln(b)\).
• The logarithm of a power is the exponent times the logarithm of the base: \(\ln(a^r) = r \cdot \ln(a)\).
• The natural logarithm of e to the power of x is x: \(\ln(e^x) = x\).
• The natural logarithm of 1 is 0: \(\ln(1) = 0\).

These properties are especially handy when simplifying expressions during differentiation and integration of functions involving e.

Composite Functions

A composite function is created when the output of one function becomes the input of another function. In mathematics, we notate a composite function as f(g(x)). Here, g is applied first, and then the result of g(x) becomes the input for the function f. When you differentiate composite functions, the chain rule is your go-to technique. It helps to separate the functions, differentiate each part with respect to its variable, then multiply the derivatives according to the chain rule formula. Understanding composite functions is essential when solving problems where variables are not just simple x or y but functions themselves, thus making the overall equation more complex.