Problem 12 - Calculus and its applications

Question

Differentiate. $$ f(x)=12^{7 x-4} $$

Step-by-Step Solution

Answer

\( f'(x) = 7 \cdot 12^{7x - 4} \cdot \ln(12) \)

Step 1: Identify the Form of the Function

Recognize that the function is in the form of an exponential function where the base is a constant and the exponent is a function of the variable. The function given is \( f(x) = 12^{7x - 4} \).

Step 2: Apply the Exponential Differentiation Rule

For a function of the form \( a^{u(x)} \), the derivative is given by \( (a^{u(x)})' = a^{u(x)} \, \ln(a) \, \cdot u'(x) \). Here, \( a = 12 \) and \( u(x) = 7x - 4 \).

Step 3: Differentiate the Exponent

Differentiate \( u(x) = 7x - 4 \) with respect to \( x \). The derivative is \( u'(x) = 7 \).

Step 4: Substitute and Simplify

Substitute \( u(x) = 7x - 4 \), \( a = 12 \), and \( u'(x) = 7 \) into the differentiation formula: \[ f'(x) = 12^{7x - 4} \cdot \ln(12) \cdot 7 \].

Step 5: Write Final Differentiated Expression

Thus, the derivative of the function \( f(x) = 12^{7x - 4} \) is given by \[ f'(x) = 7 \cdot 12^{7x - 4} \cdot \ln(12) \].

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus. It involves finding a derivative, which is a way to calculate the rate of change of a function. Essentially, when you differentiate a function, you're figuring out how that function's values change with respect to changes in its input values.

• The process begins with identifying the function you want to differentiate. The derivative is commonly represented by the symbol \( f'(x) \) or \( \frac{df}{dx} \).
• You apply rules of differentiation to discover these rate changes, which are vital in physics, engineering, economics, and several other fields for modeling and predicting behaviors.

Differentiation can be straightforward for simple functions or more complex for functions involving exponents, trigonometry, or logarithms. Mastering differentiation of all types of functions is key for anyone studying calculus.

Exponential Differentiation

Exponential differentiation is an essential skill when working with exponential functions. An exponential function is one where the variable appears in the exponent, like \( f(x) = a^{u(x)} \). The base \( a \) is a constant, but the exponent \( u(x) \) can be a function of \( x \).When differentiating exponential functions:

• The rule states that the derivative of \( a^{u(x)} \) is \( a^{u(x)} \cdot \ln(a) \cdot u'(x) \).
• You need to first differentiate the exponent \( u(x) \) to determine \( u'(x) \).
• The natural logarithm \( \ln(a) \) is involved because it is part of the rule for differentiating these kinds of functions.

By understanding exponential differentiation, you can solve problems involving exponential growth or decay, such as population growth models or radioactive decay calculations.

Calculus Techniques

Calculus techniques provide various methods and rules essential for differentiating a wide range of functions, including polynomials, exponential functions, trigonometric functions, and more.Some crucial calculus techniques include:

• **The Power Rule:** Useful for polynomials, if you have \( x^n \) as a term, its derivative is \( nx^{n-1} \).
• **The Chain Rule:** Allows you to differentiate composite functions. If a function \( y = g(f(x)) \), the derivative is \( g'(f(x)) \cdot f'(x) \).
• **The Product Rule:** Handy when dealing with products of functions. For \( y = u(x) \cdot v(x) \), the derivative \( y' = u'(x) \cdot v(x) + u(x) \cdot v'(x) \).
• **The Quotient Rule:** When dividing functions, for \( y = \frac{u(x)}{v(x)} \), its derivative \( y' = \frac{u'(x)\cdot v(x) - u(x)\cdot v'(x)}{[v(x)]^2} \).

Mastery of these techniques allows for solving complex mathematical problems and is foundational for advanced mathematics and real-world applications.