Problem 40 - Calculus Single Variable

Question

Calculate the derivative of the given xpression with respect to \(x\). $$ 10^{2 x} \cdot 2^{10 x} $$

Step-by-Step Solution

Answer

Derivative is \( f'(x) = 10^{2x} \cdot 2^{10x} (2 \ln 10 + 10 \ln 2) \).

Step 1: Identify the function and determine the rule to apply

The function is given as \( f(x) = 10^{2x} \cdot 2^{10x} \). This is a product of two functions, so we will use the product rule for differentiation, which states: \( (uv)' = u'v + uv' \).

Step 2: Differentiate the first part of the function

Let \( u = 10^{2x} \). Using the chain rule and the fact that the derivative of \( a^x \) is \( a^x \ln a \), we have:\[ u' = \frac{d}{dx}(10^{2x}) = 10^{2x} \ln 10 \cdot 2 = 2 \cdot 10^{2x} \ln 10 \].

Step 3: Differentiate the second part of the function

Now let \( v = 2^{10x} \). Again using the chain rule, we find:\[ v' = \frac{d}{dx}(2^{10x}) = 2^{10x} \ln 2 \cdot 10 = 10 \cdot 2^{10x} \ln 2 \].

Step 4: Apply the product rule

According to the product rule, the derivative of the function is:\[ f'(x) = u'v + uv' \].Substitute the derivatives computed:\[ f'(x) = (2 \cdot 10^{2x} \ln 10) \cdot 2^{10x} + 10^{2x} \cdot (10 \cdot 2^{10x} \ln 2) \].

Step 5: Simplify the expression

Simplify the expression obtained by combining like terms:\[ f'(x) = 2 \cdot 10^{2x} \cdot 2^{10x} \ln 10 + 10 \cdot 10^{2x} \cdot 2^{10x} \ln 2 \].Factor out the common factor \( 10^{2x} \cdot 2^{10x} \):\[ f'(x) = 10^{2x} \cdot 2^{10x} (2 \ln 10 + 10 \ln 2) \].

Key Concepts

Product Rule

When you have a function that is the product of two other functions, their derivative is not simply the product of their individual derivatives. The product rule helps in such situations. If we have two functions, say, \(u(x)\) and \(v(x)\), their derivative can be calculated using the product rule formula:

• \((uv)' = u'v + uv'\)

This formula states that the derivative of a product of two functions is equal to the first function \(u(x)\) times the derivative of the second function \(v(x)\), plus the second function \(v(x)\) times the derivative of the first function \(u(x)\).
For example, in the given function:

• \(f(x) = 10^{2x} \cdot 2^{10x}\)

Assign \(u = 10^{2x}\) and \(v = 2^{10x}\).
Now apply the product rule:

• Find \(u'\): the derivative of \(10^{2x}\)
• Find \(v'\): the derivative of \(2^{10x}\)
• Substitute into the product rule: \((uv)' = u'v + uv'\)

Chain Rule

The chain rule is an essential rule for finding the derivative, especially when dealing with composite functions. It facilitates the differentiation of functions whose derivatives are influenced by another function.
A composite function has the form \(f(g(x))\). The chain rule states:

• \((f(g(x)))' = f'(g(x)) \cdot g'(x)\)

Consider \(u = 10^{2x}\). Here, the outer function is \(10^y\) with the inner function \(y = 2x\). Using the chain rule:

• The derivative of \(10^y\) is \(10^y \ln 10\)
• The derivative of \(2x\) is \(2\)
• Thus, \(u' = 2 \cdot 10^{2x} \ln 10\)

Similarly, for \(v = 2^{10x}\), the outer function is \(2^y\) where \(y = 10x\). Using chain rule:

• Derivative of \(2^y\) is \(2^y \ln 2\)
• Derivative of \(10x\) is \(10\)
• So, \(v' = 10 \cdot 2^{10x} \ln 2\)

Exponential Functions

Exponential functions are significant in calculus due to their unique properties, especially when it comes to differentiation. An exponential function has the general form \(a^{f(x)}\), where \(a\) is a constant. The differentiation of exponential functions involves working out \(a^{f(x)}\) expressions.

• If \(f(x) = a^x\), the derivative is \(f'(x) = a^x \ln a\)
• This is because the rate of change of exponential functions is proportional to the function itself.

In the given exercise, you are working with exponential functions: \(10^{2x}\) and \(2^{10x}\). Here:

• For \(10^{2x}\), the derivative involves \(10^{2x} \ln 10\)
• For \(2^{10x}\), the derivative involves \(2^{10x} \ln 2\)

Understanding how to differentiate exponential expressions is crucial in many calculus problems, as exponential growth and decay models are used to describe real-world phenomena.