Differentiation and Integration?

Question

I have a couple of questions - I keep asking similar ones because, like I love that you guys can help, but no one is explaining as they go - I'm a total retard at math so I need lots of help.

Calculate the first derivative of the following:
(a) f(x) = ((e^x) - (e^-x)) / 2
(b) f(x) = ((2x - 1) / (3x + 5)
(c) f(x) = ((x^2) + (x^3))^5
(d) f(x) = sin(In((x^2)+1))

Also, find the anti-derivative of
(e) f(x) = (5x+10) / (3(x^2)) + 12x - 7)
(f) f(x) = 2(x^2) + 3x + 9

I have no idea!! TIA love tessa xxx!!

Answer 1

Seems like you need practice applying the "rules"
(a) Rule: d (e^u)/du = e^u
For this problem, the e^x term is no problem; its derivitive is e^x [which is what makes e so fascinating]. The second term is tricky. If u = -x, then du/dx = -1. By chain rule: d(f(u))/dx=
d(f(u))/du * du/dx. Then we have e^-x ( -1) .
Adding both terms together, we have 2e^x in the num. The answer is then e^x.
b) Rule: "piece by piece" rule.
If f(x) = g(x)h(x), the f'(x)= gh' + hg'
So let g= (2x-1) and h= 1/(3x+5)
dg/dx= 2 and dh/dx = -3/(3x+5)^2 [power rule]
f'(x)= -3(2x-1)/(3x+5)^2 + 2/(3x+5)
(c) Rule: Power rule and chain rule
We let u = x^2+x^3 and du/dx = 2x+3x^2
Then f(u) = u^5 and f'(x)= 5u^4 (du/dx)
Then f'(x)= 5(x^2+x^3)^4 * (2x+3x^2)

I'll skip (d). You should have the hang of it by now
Anti-derivitives (integrals). Here, there is considerable practice and inspiration and ability to do transformations. if you cant do (f), it's time to do some hard learning, because this is the easy one. The integral is 2/3 x^3 + 3/2 x^2 + 9x + coi
coi= undetermined constant of integration.

Would like to spend more time, but remember, I can get from 2 to 12 points for my efforts, and if spend 20-30 minutes doing this for each person, I either sleep very little or get very few points.

Answer 2

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