Problem 8 - Calculus Early Transcendentals

Question

Find the derivative of the following functions. $$g(x)=6 x-2 x e^{x}$$

Step-by-Step Solution

Answer

Question: Find the derivative of the function $$g(x) = 6x - 2xe^x$$. Answer: The derivative of the given function $$g(x) = 6x - 2xe^x$$ is $$g'(x) = 6 - 2e^x - 2xe^x$$.

Step 1: Differentiate the first term using the constant multiple rule

To find the derivative of $$6x$$, simply multiply the constant $$6$$ by the derivative of $$x$$, which is 1. So, the derivative of $$6x$$ is: $$6 * 1 = 6$$.

Step 2: Differentiate the second term using the product rule

Since our second term $$-2xe^x$$ is a product of two functions, we'll apply the product rule. The two functions are $$f(x) = -2x$$ and $$g(x) = e^x$$. First, find the derivative of each function: $$f'(x) = -2$$ (using the constant multiple rule) $$g'(x) = e^x$$ (derivative of exponential function) Now apply the product rule: $$(fg)'(x) = f'(x)g(x) + f(x)g'(x)$$ $$(fg)'(x) = (-2)(e^x) + (-2x)(e^x) = -2e^x - 2xe^x$$.

Step 3: Combine the derivatives

Now combine the derivatives from Steps 1 and 2 to find the overall derivative for $$g(x)$$: $$g'(x) = 6 - 2e^x - 2xe^x$$ So, the derivative of the given function $$g(x) = 6x - 2xe^x$$ is $$g'(x) = 6 - 2e^x - 2xe^x$$.

Key Concepts

Product Rule

The product rule is a fundamental tool in calculus for finding the derivative of a product of two functions. When we have a function that is the product of two separate functions, we cannot simply take the derivative of each one and multiply the results. Instead, we use the product rule.

The product rule states that if you have two functions, say \( u(x) \) and \( v(x) \), the derivative of their product \( u(x)v(x) \) is:

• \((uv)'(x) = u'(x)v(x) + u(x)v'(x)\)

This tells us to differentiate the first function and multiply it by the second, then take the derivative of the second function and multiply it by the first, then finally add those two results.

Consider a term like \(-2x e^x\) from our original function. Here, \(-2x\) is analogous to \( u(x) \) and \( e^x \) is \( v(x) \). Applying the product rule lets us effectively find the derivative by using the derivatives of \(-2x\) and \(e^x\). This concept is especially useful when dealing with complicated expressions involving standard algebraic and transcendental functions together.

Constant Multiple Rule

The constant multiple rule in calculus is straightforward yet extremely useful. It simplifies the process of differentiation when a constant is multiplied by a function. When you have a constant multiplied by a function, you can take the derivative by multiplying the constant by the derivative of the function.

In mathematical terms, if \( c \) is a constant and \( f(x) \) is a function, the derivative of \( c \times f(x) \) is:

• \(\frac{d}{dx}[cf(x)] = c \cdot \frac{d}{dx}[f(x)]\)

In our given example, to compute the derivative of \( 6x \), we identified \( 6 \) as the constant. The derivative of \( x \), which is \( 1 \), is simply multiplied by \( 6 \). Therefore, the derivative becomes \( 6 \times 1 \). The constant multiple rule streamlines finding derivatives, especially when constants are involved, making the process much faster without mistakes.

Exponential Function Derivative

The exponential function and its properties are a key part of calculus. The derivative of the function \( e^x \) is unique due to how it behaves.

The exponential function \( e^x \) remains unchanged when differentiated, meaning the derivative of \( e^x \) is, quite amazingly, just \( e^x \) itself. This is a very distinct and simple property, which makes it special among other functions, and is extremely useful in various mathematical applications.

So, each time we have \( e^x \) as part of a term, finding the derivative is straightforward. As seen in our original function \(-2xe^x\), when applying the product rule, the derivative of \( e^x \) remains \( e^x \). This streamlines computation and minimizes errors, as this property doesn't change irrespective of the other functions it’s combined with. The exponential function, due to this unique property, is prevalent in not just mathematics but also in fields such as physics, engineering, and economics.