Problem 30 - Calculus and its applications

Question

Differentiate. $$ f(x)=x^{5}-2 e^{6 x} $$

Step-by-Step Solution

Answer

The derivative is \( f'(x) = 5x^4 - 12e^{6x} \).

Step 1: Differentiate the Power Function

To differentiate the function, we split it into two constituent parts: the power function and the exponential function. Start with the power function: the derivative of \( x^5 \) with respect to \( x \) is found using the power rule \( \frac{d}{dx}x^n = nx^{n-1} \). So, the derivative of \( x^5 \) is \( 5x^4 \).

Step 2: Differentiate the Exponential Function

Now, differentiate the exponential function part \( -2e^{6x} \). The derivative of an exponential function \( e^{ax} \) is \( ae^{ax} \). Here \( a = 6 \), so differentiate to get \( -2 \times 6e^{6x} = -12e^{6x} \).

Step 3: Combine the Derivatives

Combine the derivatives from Step 1 and Step 2 to find the derivative of the whole function: \( f'(x) = 5x^4 - 12e^{6x} \). This is obtained by adding the two results because differentiation is linear.

Key Concepts

Power Rule

When it comes to differentiation, the first thing many students learn is the power rule. This rule is crucial for simplifying the process of finding derivatives of polynomial functions.
It allows us to find how the output of a function responds to changes in input, specifically when dealing with powers of a variable. The power rule states: if you have a function of the form \( f(x) = x^n \), its derivative is given by \( f'(x) = nx^{n-1} \).
This means that you multiply the original power by the coefficient (which is often 1), and then reduce the power by one.

• Example: For \( x^5 \), apply the power rule to get \( 5x^{5-1} = 5x^4 \).
• This simplifies the process of differentiation, making it just a quick computation.
• This rule only works for real numbers n.

Always remember, this rule is perfect for polynomials, sometimes saving us from more complex step-by-step calculations.

Exponential Function

Exponential functions are widespread in mathematics, especially in calculus. They have the distinctive property that the rate of growth is proportional to the current value of the function.
A common exponential function is \( e^{x} \), where \( e \) is the base of natural logarithms, approximately equal to 2.718. The wonderful property of the exponential function \( e^{x} \) is that its derivative is exactly the same as the function itself, meaning \( \frac{d}{dx} e^{x} = e^{x} \).
For more complicated exponential functions of the form \( e^{ax} \), their derivative becomes \( a \times e^{ax} \).

• For example, consider the function \( e^{6x} \). Its derivative would be \( 6 \times e^{6x} \).
• When a constant multiplier is involved, like \(-2e^{6x} \), you'd use it in the derivative calculation too: \(-2 \times 6e^{6x} = -12e^{6x} \).

This technique allows for seamless differentiation of exponential functions in calculus.

Derivative Calculation

The process of calculating derivatives is fundamental in calculus, providing valuable insights into the behavior of functions. It involves breaking down complex functions into smaller parts, differentiating each part, and then combining these results.
In the case of the function \( f(x)=x^{5}-2 e^{6 x} \), differentiation is handled by breaking it into two simpler parts: a power function and an exponential function.
We differentiate:

• The power function \( x^5 \) using the power rule, resulting in \( 5x^4 \).
• The exponential function \( -2e^{6x} \) using the properties of exponential derivatives, resulting in \(-12e^{6x} \).

Once you've found the derivatives of each component, the final step is to combine them:
\( f'(x) = 5x^4 - 12e^{6x} \).
This operation leverages the linearity of derivatives, which states that the derivative of a sum is the sum of the derivatives, allowing for easy calculation in this context.