What is the tangent of x^2e^x ?

The tangent of x^2e^x is y=(2a_{0}e^{a_{0}}+e^{a_{0}}a_{0}^2)x-a_{0}^2e^{a_{0}}-a_{0}^3e^{a_{0}}

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tangent of x^2e^x

tangent of

x^{2} e^{x}

y = (2 a_{0} e^{a_{0}} + e^{a_{0}} a_{0}^{2}) x - a_{0}^{2} e^{a_{0}} - a_{0}^{3} e^{a_{0}}

Solution steps

Compute the tangent line to the general point

x = a_{0}

Find the tangent point:

(a_{0}, a_{0}^{2} e^{a_{0}})

Find the slope of

y = x^{2} e^{x} : \frac{d y}{d x} = 2 x e^{x} + e^{x} x^{2}

Compute the slope of

y = x^{2} e^{x}

x = a_{0} : m = 2 a_{0} e^{a_{0}} + e^{a_{0}} a_{0}^{2}

Find the line with slope m=

2 a_{0} e^{a_{0}} + e^{a_{0}} a_{0}^{2}

and passing through

(a_{0}, a_{0}^{2} e^{a_{0}}) : y = (2 a_{0} e^{a_{0}} + e^{a_{0}} a_{0}^{2}) x - a_{0}^{2} e^{a_{0}} - a_{0}^{3} e^{a_{0}}

y = (2 a_{0} e^{a_{0}} + e^{a_{0}} a_{0}^{2}) x - a_{0}^{2} e^{a_{0}} - a_{0}^{3} e^{a_{0}}

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What is the tangent of x^2e^x ?
The tangent of x^2e^x is y=(2a_{0}e^{a_{0}}+e^{a_{0}}a_{0}^2)x-a_{0}^2e^{a_{0}}-a_{0}^3e^{a_{0}}