Find the Second Derivative 3x^2e^(2x)

$3 x^{2} e^{2 x}$

Step 1

Find the first derivative.

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{2} e^{2 x}$ with respect to $x$ is $3 \frac{d}{d x} [x^{2} e^{2 x}]$ .

$f' (x) = 3 \frac{d}{d x} (x^{2} e^{2 x})$

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

$f' (x) = 3 (x^{2} \frac{d}{d x} (e^{2 x}) + e^{2 x} \frac{d}{d x} (x^{2}))$

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

To apply the Chain Rule, set $u$ as $2 x$ .

$f' (x) = 3 (x^{2} (\frac{d}{d u} (e^{u}) \frac{d}{d x} (2 x)) + e^{2 x} \frac{d}{d x} (x^{2}))$

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

$f' (x) = 3 (x^{2} (e^{u} \frac{d}{d x} (2 x)) + e^{2 x} \frac{d}{d x} (x^{2}))$

Replace all occurrences of $u$ with $2 x$ .

$f' (x) = 3 (x^{2} (e^{2 x} \frac{d}{d x} (2 x)) + e^{2 x} \frac{d}{d x} (x^{2}))$

Differentiate.

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$f' (x) = 3 (x^{2} (e^{2 x} (2 \frac{d}{d x} (x))) + e^{2 x} \frac{d}{d x} (x^{2}))$

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

$f' (x) = 3 (x^{2} (e^{2 x} (2 \cdot 1)) + e^{2 x} \frac{d}{d x} (x^{2}))$

Simplify the expression.

Multiply $2$ by $1$ .

$f' (x) = 3 (x^{2} (e^{2 x} \cdot 2) + e^{2 x} \frac{d}{d x} (x^{2}))$

Move $2$ to the left of $e^{2 x}$ .

$f' (x) = 3 (x^{2} (2 \cdot e^{2 x}) + e^{2 x} \frac{d}{d x} (x^{2}))$

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

$f' (x) = 3 (x^{2} (2 e^{2 x}) + e^{2 x} (2 x))$

Simplify.

Apply the distributive property.

$f' (x) = 3 (x^{2} (2 e^{2 x})) + 3 (e^{2 x} (2 x))$

Combine terms.

Multiply $2$ by $3$ .

$f' (x) = 6 (x^{2} (e^{2 x})) + 3 (e^{2 x} (2 x))$

Multiply $2$ by $3$ .

$f' (x) = 6 x^{2} e^{2 x} + 6 e^{2 x} x$

Reorder terms.

$f' (x) = 6 e^{2 x} x^{2} + 6 e^{2 x} x$

Reorder factors in $6 e^{2 x} x^{2} + 6 e^{2 x} x$ .

$f' (x) = 6 x^{2} e^{2 x} + 6 x e^{2 x}$

Step 2

Find the second derivative.

By the Sum Rule, the derivative of $6 x^{2} e^{2 x} + 6 x e^{2 x}$ with respect to $x$ is $\frac{d}{d x} [6 x^{2} e^{2 x}] + \frac{d}{d x} [6 x e^{2 x}]$ .

$f'' (x) = \frac{d}{d x} (6 x^{2} e^{2 x}) + \frac{d}{d x} (6 x e^{2 x})$

Evaluate $\frac{d}{d x} [6 x^{2} e^{2 x}]$ .

Since $6$ is constant with respect to $x$ , the derivative of $6 x^{2} e^{2 x}$ with respect to $x$ is $6 \frac{d}{d x} [x^{2} e^{2 x}]$ .

$f'' (x) = 6 \frac{d}{d x} (x^{2} e^{2 x}) + \frac{d}{d x} (6 x e^{2 x})$

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

$f'' (x) = 6 (x^{2} \frac{d}{d x} (e^{2 x}) + e^{2 x} \frac{d}{d x} (x^{2})) + \frac{d}{d x} (6 x e^{2 x})$

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

To apply the Chain Rule, set $u_{1}$ as $2 x$ .

$f'' (x) = 6 (x^{2} (\frac{d}{d u (1)} (e^{u_{1}}) \frac{d}{d x} (2 x)) + e^{2 x} \frac{d}{d x} (x^{2})) + \frac{d}{d x} (6 x e^{2 x})$

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

$f'' (x) = 6 (x^{2} (e^{u_{1}} \frac{d}{d x} (2 x)) + e^{2 x} \frac{d}{d x} (x^{2})) + \frac{d}{d x} (6 x e^{2 x})$

Replace all occurrences of $u_{1}$ with $2 x$ .

$f'' (x) = 6 (x^{2} (e^{2 x} \frac{d}{d x} (2 x)) + e^{2 x} \frac{d}{d x} (x^{2})) + \frac{d}{d x} (6 x e^{2 x})$

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$f'' (x) = 6 (x^{2} (e^{2 x} (2 \frac{d}{d x} (x))) + e^{2 x} \frac{d}{d x} (x^{2})) + \frac{d}{d x} (6 x e^{2 x})$

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

$f'' (x) = 6 (x^{2} (e^{2 x} (2 \cdot 1)) + e^{2 x} \frac{d}{d x} (x^{2})) + \frac{d}{d x} (6 x e^{2 x})$

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

$f'' (x) = 6 (x^{2} (e^{2 x} (2 \cdot 1)) + e^{2 x} (2 x)) + \frac{d}{d x} (6 x e^{2 x})$

Multiply $2$ by $1$ .

$f'' (x) = 6 (x^{2} (e^{2 x} \cdot 2) + e^{2 x} (2 x)) + \frac{d}{d x} (6 x e^{2 x})$

Move $2$ to the left of $e^{2 x}$ .

$f'' (x) = 6 (x^{2} (2 e^{2 x}) + e^{2 x} (2 x)) + \frac{d}{d x} (6 x e^{2 x})$

Evaluate $\frac{d}{d x} [6 x e^{2 x}]$ .

Since $6$ is constant with respect to $x$ , the derivative of $6 x e^{2 x}$ with respect to $x$ is $6 \frac{d}{d x} [x e^{2 x}]$ .

$f'' (x) = 6 (x^{2} (2 e^{2 x}) + e^{2 x} (2 x)) + 6 \frac{d}{d x} (x e^{2 x})$

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

$f'' (x) = 6 (x^{2} (2 e^{2 x}) + e^{2 x} (2 x)) + 6 (x \frac{d}{d x} (e^{2 x}) + e^{2 x} \frac{d}{d x} (x))$

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

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

$f'' (x) = 6 (x^{2} (2 e^{2 x}) + e^{2 x} (2 x)) + 6 (x (\frac{d}{d u (2)} (e^{u_{2}}) \frac{d}{d x} (2 x)) + e^{2 x} \frac{d}{d x} (x))$

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{2}} [a^{u_{2}}]$ is $a^{u_{2}} \ln (a)$ where $a$ = $e$ .

$f'' (x) = 6 (x^{2} (2 e^{2 x}) + e^{2 x} (2 x)) + 6 (x (e^{u_{2}} \frac{d}{d x} (2 x)) + e^{2 x} \frac{d}{d x} (x))$

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

$f'' (x) = 6 (x^{2} (2 e^{2 x}) + e^{2 x} (2 x)) + 6 (x (e^{2 x} \frac{d}{d x} (2 x)) + e^{2 x} \frac{d}{d x} (x))$

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$f'' (x) = 6 (x^{2} (2 e^{2 x}) + e^{2 x} (2 x)) + 6 (x (e^{2 x} (2 \frac{d}{d x} (x))) + e^{2 x} \frac{d}{d x} (x))$

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

$f'' (x) = 6 (x^{2} (2 e^{2 x}) + e^{2 x} (2 x)) + 6 (x (e^{2 x} (2 \cdot 1)) + e^{2 x} \frac{d}{d x} (x))$

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

$f'' (x) = 6 (x^{2} (2 e^{2 x}) + e^{2 x} (2 x)) + 6 (x (e^{2 x} (2 \cdot 1)) + e^{2 x} \cdot 1)$

Multiply $2$ by $1$ .

$f'' (x) = 6 (x^{2} (2 e^{2 x}) + e^{2 x} (2 x)) + 6 (x (e^{2 x} \cdot 2) + e^{2 x} \cdot 1)$

Move $2$ to the left of $e^{2 x}$ .

$f'' (x) = 6 (x^{2} (2 e^{2 x}) + e^{2 x} (2 x)) + 6 (x (2 \cdot e^{2 x}) + e^{2 x} \cdot 1)$

Multiply $e^{2 x}$ by $1$ .

$f'' (x) = 6 (x^{2} (2 e^{2 x}) + e^{2 x} (2 x)) + 6 (x (2 e^{2 x}) + e^{2 x})$

Simplify.

Apply the distributive property.

$f'' (x) = 6 (x^{2} (2 e^{2 x})) + 6 (e^{2 x} (2 x)) + 6 (x (2 e^{2 x}) + e^{2 x})$

Apply the distributive property.

$f'' (x) = 6 (x^{2} (2 e^{2 x})) + 6 (e^{2 x} (2 x)) + 6 (x (2 e^{2 x})) + 6 e^{2 x}$

Combine terms.

Multiply $2$ by $6$ .

$f'' (x) = 12 (x^{2} (e^{2 x})) + 6 (e^{2 x} (2 x)) + 6 (x (2 e^{2 x})) + 6 e^{2 x}$

Multiply $2$ by $6$ .

$f'' (x) = 12 x^{2} e^{2 x} + 12 (e^{2 x} (x)) + 6 (x (2 e^{2 x})) + 6 e^{2 x}$

Multiply $2$ by $6$ .

$f'' (x) = 12 x^{2} e^{2 x} + 12 e^{2 x} x + 12 (x (e^{2 x})) + 6 e^{2 x}$

Add $12 e^{2 x} x$ and $12 x e^{2 x}$ .

Move $e^{2 x}$ .

$f'' (x) = 12 x^{2} e^{2 x} + 12 x e^{2 x} + 12 x e^{2 x} + 6 e^{2 x}$

Add $12 x e^{2 x}$ and $12 x e^{2 x}$ .

$f'' (x) = 12 x^{2} e^{2 x} + 24 x e^{2 x} + 6 e^{2 x}$

Reorder terms.

$f'' (x) = 12 e^{2 x} x^{2} + 24 e^{2 x} x + 6 e^{2 x}$

Reorder factors in $12 e^{2 x} x^{2} + 24 e^{2 x} x + 6 e^{2 x}$ .

$f'' (x) = 12 x^{2} e^{2 x} + 24 x e^{2 x} + 6 e^{2 x}$

Find the Second Derivative 3x^2e^(2x)

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