(d^2)/(dx^2)(3e^{x^2}) — step-by-step solution

⚠Ⓘ 3e is read as the product 3*e: for a power, write 3^e‖(d^2)/(dx^2)(3e^{x^2}) is read as double derivative of (3e^{x^2}) for x: use dif(expression, variable) for a derivative with respect to that variable, use (numerator)/(denominator) for a quotient‖(3e^{x^2}) is read as (3e^(x^2))

⇥1) Let's perform the differentiation:

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

⇥We differentiate using:

\frac{d}{d x} (\exp (f)) = \frac{d}{d x} (f) \cdot \exp (f)

(where f =

x^{2}

)

⇥

3 \cdot \frac{d}{d x} (x^{2}) \cdot e^{x^{2}}

⇥

3 \cdot 2 x \cdot e^{x^{2}}

⇥

6 x e^{x^{2}}

2) Let's differentiate:

\frac{d}{d x} (6 e^{x^{2}} x)

6 \cdot \frac{d}{d x} (e^{x^{2}} x)

For

U1

e^{x^{2}}

and

V2

x

, use the product/quotient rule:

\frac{d}{d x} (U1 \cdot V2) = \frac{d}{d x} (U1) \cdot V2 + U1 \cdot \frac{d}{d x} (V2)

3) Let's differentiate

U1

\frac{d}{d x} (e^{x^{2}})

We differentiate using:

\frac{d}{d x} (\exp (f)) = \frac{d}{d x} (f) \cdot \exp (f)

(where f =

x^{2}

)

\frac{d}{d x} (x^{2}) \cdot e^{x^{2}}

2 x \cdot e^{x^{2}}

4) Let's differentiate

V2

\frac{d}{d x} (x)

1

5) You can evaluate the resulted derivative (

U1 \cdot V2

)':

2 x e^{x^{2}} \cdot x + e^{x^{2}} \cdot 1

2 x^{2} e^{x^{2}} + e^{x^{2}}

6) You can then resume the derivation:

6 \cdot (2 x^{2} e^{x^{2}} + e^{x^{2}})

12 x^{2} e^{x^{2}} + 6 e^{x^{2}}

▶Derivative:

12 e^{x^{2}} x^{2} + 6 e^{x^{2}}