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Chapter 4: Problem 40

Find the slope of the tangent line to the curve \(y=x e^{x}\) at \((1, e).\)

Short Answer

Expert verified
The slope is \(2e\).

Step by step solution

01

- Recall the formula for the slope of the tangent line

The slope of the tangent line to a curve at a given point is given by the derivative of the function evaluated at that point.
02

- Find the derivative of the function

Given the function is \(y = x e^{x}\), use the product rule for differentiation. Let \(u = x\) and \(v = e^{x}\). The product rule states that \((uv)' = u'v + uv'\). Hence,\[ \frac{dy}{dx} = \frac{d}{dx} (x e^x) = \frac{d}{dx}(x) \times e^x + x \times \frac{d}{dx}(e^x) \]Now calculate the derivatives: \( \frac{d}{dx}(x) = 1 \) and \( \frac{d}{dx}(e^x) = e^x \). Substituting these in:\[ \frac{dy}{dx} = 1 \times e^x + x \times e^x = e^x (1 + x) \]
03

- Evaluate the derivative at the given point

To find the slope of the tangent line at the point \((1, e)\), substitute \(x = 1\) into the derivative. Thus,\[ \frac{dy}{dx} \bigg|_{x=1} = e^1 (1 + 1) = e \times 2 = 2e \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Derivative Calculation
The derivative measures how a function changes as its input changes. It gives us the slope of the tangent line to the curve of the function at any given point. For the function given in our exercise, which is y = x e^{x}, we need to find the derivative in order to determine the slope of the tangent line.

  • Start by identifying the function: y = x e^{x}.

  • To find the derivative, apply the rules of differentiation.

In our context, differentiating y with respect to x involves using the product rule (explained in the next section). The final expression for the derivative we achieve is:

\[ \frac{dy}{dx} = e^x (1 + x) \]
This function gives us the slope of the tangent line at any x value on the curve.
Product Rule
The product rule is a formula used to find the derivative of functions that are products of two other functions. When we have a function that can be written as the product of two parts, the derivative is calculated in a specific way.

Suppose we have two functions: u(x) and v(x). If y is the product of these functions y = u(x) \times v(x), then the product rule states:

\[ (uv)' = u'v + uv' \]
Let's apply this to our function y = x e^{x}. Here, we can identify:

  • u = x

  • v = e^x

Next, calculate the derivatives of u and v:
  • u' = x \rightarrow u' = 1

  • v' = e^x \rightarrow v' = e^x

Substitute these back into the product rule formula:

\[ \frac{dy}{dx} = u'v + uv' = 1 \times e^x + x \times e^x = e^x (1 + x) \]
Hence, we've used the product rule to find the derivative of y = x e^{x}.
Curve Evaluation
Finally, after finding the derivative of our function, we can evaluate the derivative at a specific point to determine the slope of the tangent line at that point. In our exercise, we're asked to find the slope of the tangent at the point (1, e).

  • The derivative of our function is e^x (1 + x).

  • To evaluate this at x = 1, we substitute 1 for x in the derivative.

This gives us:

\[ \frac{dy}{dx} \bigg|_{x=1} = e^1 (1 + 1) = e \times 2 = 2e \]
Therefore, at the point (1, e), the slope of the tangent line to the curve y = x e^{x} is 2e.

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Most popular questions from this chapter

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