$\text { Differentiate the function } f ( x ) = 5 e ^ { 3 x ^ { 2 } }$
A)$30 x e ^ { 6 x }$
B) $15 x e ^ { 3 x ^ { 2 } }$
C) $15 e ^ { 3 x ^ { 2 } }$
D) $30 x e ^ { 3 x ^ { 2 } }$
E) $15 e ^ { 6 x }$

Question

$\text { Differentiate the function } f ( x ) = 5 e ^ { 3 x ^ { 2 } }$ 
A)$30 x e ^ { 6 x }$
B) $15 x e ^ { 3 x ^ { 2 } }$
C) $15 e ^ { 3 x ^ { 2 } }$
D) $30 x e ^ { 3 x ^ { 2 } }$
E) $15 e ^ { 6 x }$

Quizplus · Accepted Answer

To differentiate a function in the form of $f(x)=ae^{g(x)}$, we use the chain rule. The derivative is given by:
$$\frac{d}{dx}\left(ae^{g(x)}\right) = ae^{g(x)}\cdot g'(x)$$
In this case, we have $a=5$ and $g(x)=3x^2$. Taking the derivative of $g(x)$ with respect to $x$ gives:
$$g'(x) = 6x$$
Therefore, the derivative of $f(x)$ with respect to $x$ is:
$$\frac{d}{dx}\left(5e^{3x^2}\right) = 5e^{3x^2}\cdot6x = \boxed{30xe^{3x^2}}$$

Differentiate the function f(x)=5e3x2\text { Differentiate the function } f ( x ) = 5 e ^ { 3 x ^ { 2 } } Differentiate the function f(x)=5e3x2

$\text { Differentiate the function } f ( x ) = 5 e ^ { 3 x ^ { 2 } }$