How do you find the equation of the tangent line to the graph #y=x^2e^x2xe^x+2e^x# through point (1,e)?
Slope=
You differentiate of course.
Thus, the equation is simplified to
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The slope of the tangent is
The equation to the tangent at P is
graph{(e^x(x^22x+2)y)(ye^1 x)((x1)^2+(y2.73)^2.0007)=0 [1, 2, 1.5, 4]}
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To find the equation of the tangent line to the graph of the function y = x^2e^x  2xe^x + 2e^x through the point (1, e), we need to follow these steps:
 Find the derivative of the function y = x^2e^x  2xe^x + 2e^x using the product rule and the chain rule.
 Substitute the xcoordinate of the given point (1, e) into the derivative to find the slope of the tangent line at that point.
 Use the pointslope form of a linear equation, y  y1 = m(x  x1), where (x1, y1) is the given point and m is the slope, to write the equation of the tangent line.
Let's go through these steps:

Differentiating the function y = x^2e^x  2xe^x + 2e^x:
 Apply the product rule: (uv)' = u'v + uv'
 Apply the chain rule: (e^x)' = e^x
 Differentiate each term:
 (x^2e^x)' = 2xe^x + x^2e^x
 (2xe^x)' = 2e^x  2xe^x
 (2e^x)' = 2e^x
 Combine the derivatives: y' = (2xe^x + x^2e^x)  (2e^x + 2xe^x) + 2e^x
 Simplify: y' = x^2e^x

Substitute x = 1 into the derivative to find the slope at the point (1, e):
 y' = (1^2)(e^1) = e

Use the pointslope form with the given point (1, e) and slope e:
 y  e = e(x  1)
 Simplify: y  e = ex  e
 Rearrange: y = ex  e + e
 Final equation of the tangent line: y = ex
Therefore, the equation of the tangent line to the graph y = x^2e^x  2xe^x + 2e^x through the point (1, e) is y = ex.
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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