What is the derivative of (x^2+x^3e^{-1/x}) ?

The derivative of (x^2+x^3e^{-1/x}) is 3e^{-1/x}x^2+3e^{-1/x}x+e^{-1/x}

What is the first derivative of (x^2+x^3e^{-1/x}) ?

The first derivative of (x^2+x^3e^{-1/x}) is 3e^{-1/x}x^2+3e^{-1/x}x+e^{-1/x}

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\frac{d}{d x} ((x^{2} + x^{3}) e^{- \frac{1}{x}})

3 e^{- \frac{1}{x}} x^{2} + 3 e^{- \frac{1}{x}} x + e^{- \frac{1}{x}}

Solution steps

\frac{d}{d x} ((x^{2} + x^{3}) e^{- \frac{1}{x}})

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x^{2} + x^{3}, g = e^{- \frac{1}{x}}

= \frac{d}{d x} (x^{2} + x^{3}) e^{- \frac{1}{x}} + \frac{d}{d x} (e^{- \frac{1}{x}}) (x^{2} + x^{3})

\frac{d}{d x} (x^{2} + x^{3}) = 2 x + 3 x^{2}

\frac{d}{d x} (e^{- \frac{1}{x}}) = \frac{e ^{- \frac{1}{x}}}{x ^{2}}

= (2 x + 3 x^{2}) e^{- \frac{1}{x}} + \frac{e ^{- \frac{1}{x}}}{x ^{2}} (x^{2} + x^{3})

Simplify

(2 x + 3 x^{2}) e^{- \frac{1}{x}} + \frac{e ^{- \frac{1}{x}}}{x ^{2}} (x^{2} + x^{3}) : 3 e^{- \frac{1}{x}} x^{2} + 3 e^{- \frac{1}{x}} x + e^{- \frac{1}{x}}

= 3 e^{- \frac{1}{x}} x^{2} + 3 e^{- \frac{1}{x}} x + e^{- \frac{1}{x}}

What is the derivative of (x^2+x^3e^{-1/x}) ?
The derivative of (x^2+x^3e^{-1/x}) is 3e^{-1/x}x^2+3e^{-1/x}x+e^{-1/x}
What is the first derivative of (x^2+x^3e^{-1/x}) ?
The first derivative of (x^2+x^3e^{-1/x}) is 3e^{-1/x}x^2+3e^{-1/x}x+e^{-1/x}