Find dx/dy y=x^2e^x

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

Step 1

Differentiate both sides of the equation.

$\frac{d}{d y} (y) = \frac{d}{d y} (x^{2} e^{x})$

Step 2

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = 1$ .

$1$

Step 3

Differentiate the right side of the equation.

Differentiate using the Product Rule which states that $\frac{d}{d y} [f (y) g (y)]$ is $f (y) \frac{d}{d y} [g (y)] + g (y) \frac{d}{d y} [f (y)]$ where $f (y) = x^{2}$ and $g (y) = e^{x}$ .

$x^{2} \frac{d}{d y} [e^{x}] + e^{x} \frac{d}{d y} [x^{2}]$

Differentiate using the chain rule, which states that $\frac{d}{d y} [f (g (y))]$ is $f' (g (y)) g' (y)$ where $f (y) = e^{y}$ and $g (y) = x$ .

To apply the Chain Rule, set $u_{1}$ as $x$ .

$x^{2} (\frac{d}{d u_{1}} [e^{u_{1}}] \frac{d}{d y} [x]) + e^{x} \frac{d}{d y} [x^{2}]$

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $e$ .

$x^{2} (e^{u_{1}} \frac{d}{d y} [x]) + e^{x} \frac{d}{d y} [x^{2}]$

Replace all occurrences of $u_{1}$ with $x$ .

$x^{2} (e^{x} \frac{d}{d y} [x]) + e^{x} \frac{d}{d y} [x^{2}]$

Rewrite $\frac{d}{d y} [x]$ as $x'$ .

$x^{2} (e^{x} x') + e^{x} \frac{d}{d y} [x^{2}]$

Differentiate using the chain rule, which states that $\frac{d}{d y} [f (g (y))]$ is $f' (g (y)) g' (y)$ where $f (y) = y^{2}$ and $g (y) = x$ .

To apply the Chain Rule, set $u_{2}$ as $x$ .

$x^{2} (e^{x} x') + e^{x} (\frac{d}{d u_{2}} [{u_{2}}^{2}] \frac{d}{d y} [x])$

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 2$ .

$x^{2} (e^{x} x') + e^{x} (2 u_{2} \frac{d}{d y} [x])$

Replace all occurrences of $u_{2}$ with $x$ .

$x^{2} (e^{x} x') + e^{x} (2 x \frac{d}{d y} [x])$

Rewrite $\frac{d}{d y} [x]$ as $x'$ .

$x^{2} (e^{x} x') + e^{x} (2 x x')$

Simplify.

Reorder terms.

$e^{x} x^{2} x' + 2 e^{x} x x'$

Reorder factors in $e^{x} x^{2} x' + 2 e^{x} x x'$ .

$x^{2} e^{x} x' + 2 x e^{x} x'$

Step 4

Reform the equation by setting the left side equal to the right side.

$1 = x^{2} e^{x} x' + 2 x e^{x} x'$

Step 5

Solve for $x'$ .

Rewrite the equation as $x^{2} e^{x} x' + 2 x e^{x} x' = 1$ .

$x^{2} e^{x} x' + 2 x e^{x} x' = 1$

Reorder factors in $x^{2} e^{x} x' + 2 x e^{x} x'$ .

$x^{2} x' e^{x} + 2 x x' e^{x} = 1$

Factor $x x' e^{x}$ out of $x^{2} x' e^{x} + 2 x x' e^{x}$ .

Factor $x x' e^{x}$ out of $x^{2} x' e^{x}$ .

$x x' e^{x} x + 2 x x' e^{x} = 1$

Factor $x x' e^{x}$ out of $2 x x' e^{x}$ .

$x x' e^{x} x + x x' e^{x} \cdot 2 = 1$

Factor $x x' e^{x}$ out of $x x' e^{x} x + x x' e^{x} \cdot 2$ .

$x x' e^{x} (x + 2) = 1$

Divide each term in $x x' e^{x} (x + 2) = 1$ by $x e^{x} (x + 2)$ and simplify.

Divide each term in $x x' e^{x} (x + 2) = 1$ by $x e^{x} (x + 2)$ .

$\frac{x x' e^{x} (x + 2)}{x e^{x} (x + 2)} = \frac{1}{x e^{x} (x + 2)}$

Simplify the left side.

Cancel the common factor of $x$ .

Cancel the common factor.

$\frac{x x' e^{x} (x + 2)}{x e^{x} (x + 2)} = \frac{1}{x e^{x} (x + 2)}$

Rewrite the expression.

$\frac{x' e^{x} (x + 2)}{e^{x} (x + 2)} = \frac{1}{x e^{x} (x + 2)}$

Cancel the common factor of $e^{x}$ .

Cancel the common factor.

$\frac{x' e^{x} (x + 2)}{e^{x} (x + 2)} = \frac{1}{x e^{x} (x + 2)}$

Rewrite the expression.

$\frac{x' (x + 2)}{x + 2} = \frac{1}{x e^{x} (x + 2)}$

Cancel the common factor of $x + 2$ .

Cancel the common factor.

$\frac{x' (x + 2)}{x + 2} = \frac{1}{x e^{x} (x + 2)}$

Divide $x'$ by $1$ .

$x' = \frac{1}{x e^{x} (x + 2)}$

Step 6

Replace $x'$ with $\frac{d x}{d y}$ .

$\frac{d x}{d y} = \frac{1}{x e^{x} (x + 2)}$

Find dx/dy y=x^2e^x

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