Find the slope of the graph of f(x) = 2 + xe^x at the poi...

Question

Find the slope of the graph of f(x) = 2 + xe^x at the point (0, 2).

Accepted Answer

Welcome back, everyone. What is the slope of the tangent line to F of X equals 3 + 2 X eats the power of X at a.0.3. We are given four answer choices A says 0, B1, C2 and D3. So let's recall that the slope of a tangent line is simply the derivative of the original function. So F. And if we're looking at a specific point, we can say that F at a specific point X not. So we need to find two parts. Number one, we want to differentiate the differentiate the parent functions, so we want to identify FX, and then our second step is to evaluate this derivative. At the given point X not in this case, the point of interest is 0, so we don't really need to use the y value because by definition the slope of the tangent line is F at point X0. In this case, X 0 is 0, so we will evaluate the derivative at 0.0. So we can summarize the goal of this problem by simply saying that we are evaluating the derivative at 0.0. Let's begin with the first step. We want to evaluate the derivative of F of X, which is the derivative of 3 + 2 X eats the power of X. And now this can be written as the derivative of 3 + 2 multiplied by the derivative of X, multiplied by the power of X. Now first of all, the derivative of 3 is 0 because it is a constant. So we simply want to apply the product rule or X multiplied by e to the power of X, and the product rule says that if we are multiplying F by G and finding the derivative of this function, this is F G. Plus FG. So, in this case, if we use different notations, I suppose that instead of F, we simply choose H to avoid any kind of confusion. We're going to replace F with H, so H multiplied by G derivative is HG plus Hg. In this case, we can set H equals x our first function, and G equals eats the power of x our second function. So H is 1 based on the power rule and G is eats the power of x. Based on the table of basic derivatives. So we are going to get two ins. First of all, we take H, which is 1 multiplied by G, which is E to the power of X, plus H, which is X, multiplied by G which is the power of X. And now let's simplify, we are going to get 2. We can factor out E to the power of X, and in parties we're going to get 1 + X. So we have F and now our goal is to evaluate F at 0.0. So we're subsiding 0 for every X and we get 2 Es, the power of 0, multiplied by 1 + 0. So essentially we have 2 multiplied by 1, because it's the power of 0 is 1, multiplied by 1, which gives us 2. And our final answer to this problem corresponds to the answer choice C. Thank you for watching.

Find the slope of the graph of f(x) = 2 + xe^x at the point (0, 2).

Key Concepts

Derivative

Exponential Function

Point-Slope Form

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