Question

Find the derivatives of the functions defined as follows:
y= -3e^(3x^2 +5)

Answer 1

\[\Large\bf\sf y\quad=\quad -3e^{3x^2+5}\]Hmm ok so to start...
Do you recall the derivative of:\[\Large\bf\sf e^x\]And do you also know how to find the derivative of this?\[\Large\bf\sf 2e^x\]

Answer 2

e^x is just e^x right?

Answer 3

and 2e^x is just 1

Answer 4

e^x is just e^x, good.\[\Large\bf\sf (2e^x)'\quad=\quad 2(e^x)'\quad=\quad 2e^x\]Constant coefficients don't affect the differentiation process.
We can ignore them when taking a derivative.
See how I pulled the 2 outside and then took the derivative of e^x as we normally would?

Answer 5

Ohhh right. Okay!

Yeah, gotcha.

Answer 6

Same idea with our problem, we'll just need to remember to apply the chain rule.\[\Large\bf\sf \left(-3e^{stuff}\right)'\quad=\quad -3\left(e^{stuff}\right)'\quad=\quad -3e^{stuff}(stuff)'\]

Answer 7

The exponential is giving us the same thing back.
The chain rule rule tells us to multiply by the derivative of the inner function.

Answer 8

Okay!

Answer 9

\[\Large\bf\sf \left(-3e^{3x^2+5}\right)'\quad=\quad -3e^{3x^2+5}(3x^2+5)'\]

Answer 10

So what do we get from our chain? :)