How Do You Simplify the Derivative Expression of \( y = e^{7x+4} \)?

[Chadlee88]

Could som1 please tell me what the next steps would be to be able to remove the h in the denomenator.

y = e^(7x+4)


Definition: lim f(x+h) - f(x)
h->0 h

lim (e^(7(x+h) + 4) - (e^(7x+4))
h->0 h

lim (e^(7x + 7h + 4)) - (e^(7x +4))
h->0 h

 

[HallsofIvy]

How about using some of the properties of the exponential?
$$e^{7x+7h+4}= e^{7h}e^{7x+ 4}$$
(yes, I could also have separated the "4" but it is the "h" that is important)
so
$$e^{7x+ 7h+ 4}- e^{7x+ 4}= e^{7x+4}(e^{7h}- 1)$$
You will still have to deal with
$$\lim_{h\rightarrow 0}\frac{e^{7h}-1}{h}= 7\lim_{h\rightarrow 0}\frac{e^{7h}-1}{7h}$$
and, taking k= 7h,
$$7\lim_{k\rightarrow 0}\frac{e^{k}-1}{k}$$

but if you know how to deal with the derivative of e^x you should be able to do that.

 

[Chadlee88]

i know that the final answer is 7 x .5 = 3.5 but i don't get how you got rid of e^(7x+4).

e^(7x+7h+4) - e^(7x+4) = e^(7x+4)(e^(7h) - 1) and then somehow
the e^(7x + 4) disappears and u get lim e^(7h-1)
h->0 h

 

[e(ho0n3]

Do you know how to derive the derivative of e^x from the definition? If you can't, then you won't be able to solve this problem.

 

[HallsofIvy]

i know that the final answer is 7 x .5 = 3.5 but i don't get how you got rid of e^(7x+4).

e^(7x+7h+4) - e^(7x+4) = e^(7x+4)(e^(7h) - 1) and then somehow
the e^(7x + 4) disappears and u get lim e^(7h-1)
h->0 h

If that is the answer, then what is the question?

The derivative of e^7x+4 is 7e^7x+4! You don't "rid of" e^7x+4, it's part of the answer. Since you assert that the answer is a number, 3.5, is it possible that the problem asks for the derivative at a given value of x?