Problem 11 - Calculus and its applications

Question

Differentiate. $$ f(x)=3^{x^{4}+1} $$

Step-by-Step Solution

Answer

The derivative is \( f'(x) = 4x^3 \ln(3) \cdot 3^{x^4 + 1} \).

Step 1: Identify the Differentiation Technique

Notice that the function \( f(x) = 3^{x^4 + 1} \) is an exponential function where the base is a constant \( 3 \), and the exponent is the expression \( x^4 + 1 \). Therefore, we'll use the chain rule along with the derivative of an exponential function with a constant base.

Step 2: Apply the Exponential Derivative Rule

The general derivative of an exponential function with base 'a' is \( \frac{d}{dx} a^u = a^u \ln(a) \frac{du}{dx} \), where \( u \) is a function of \( x \). In this case, \( a = 3 \) and \( u = x^4 + 1 \).

Step 3: Differentiate the Exponent

Compute the derivative of the exponent \( u = x^4 + 1 \). The derivative \( \frac{du}{dx} = 4x^3 \), since the derivative of \( x^4 \) is \( 4x^3 \) and the derivative of the constant \( 1 \) is \( 0 \).

Step 4: Combine Results Using the Chain Rule

Substitute back into the derivative formula: \[ f'(x) = 3^{x^4 + 1} \ln(3) \cdot (4x^3) \]. This combines the derivative of the exponent from Step 3 with the logarithmic term from Step 2.

Step 5: Simplify the Expression

Simplify the expression obtained: \[ f'(x) = 4x^3 \ln(3) \cdot 3^{x^4 + 1} \]. This form makes it clear that the differentiation has been completed.

Key Concepts

Exponential Functions

Exponential functions are a type of function where a constant base is raised to a variable exponent. This structure is quite unique and powerful in mathematics. They exhibit rapid growth or decay due to their nature.
Imagine something that doubles every time you look at it, that's how exponential functions can behave!
Here are some characteristics of exponential functions:

• The base of the exponential function, like the number "3" in our function, is a positive constant.


• The exponent is a variable that often contains other expressions, such as polynomials, like our "x^4 + 1."


• Graphically, exponential functions have a specific curve that starts slow and then rises or falls quickly.

Understanding exponential functions is vital as they form the backbone for concepts like compound interest and population growth in real-world scenarios.

Chain Rule

The chain rule is an essential differentiation tool in calculus. It lets us take derivatives of composite functions, meaning functions within functions!
Think of it as peeling back layers of an onion to find the derivative inside.
Here's how you can use the chain rule effectively:

• Identify the outer function and the inner function. For our function, the outer function is the exponential, while the inner function is the polynomial "x^4 + 1."


• Find the derivative of the outer function while keeping the inner function the same. Then, multiply it by the derivative of the inner function.

This method helps to address problems where there are nested functions, making differentiation much more manageable.

Differentiation Techniques

Differentiation involves finding the rate at which a function changes at any given point. It's a core concept of calculus that is primarily about finding derivatives.
There are several differentiation techniques, and choosing the right one can make all the difference:

• For simple polynomials, apply the power rule, which involves multiplying the exponent by the coefficient and decreasing the exponent by one.


• When dealing with exponential functions, like in our problem, use the derivative rule for exponential functions combined with techniques like the chain rule to handle composite expressions.

Mastering these techniques is crucial since they can be combined to solve complicated problems efficiently. Practice with different types of functions to strengthen understanding and application of these techniques.