!im a rising senior/new to calculus/starting to learn before school starts/plz help!!

[swamanator]

ok so i am new to calculus ...actully havnt even started to learn it....to make life easier im starting before school starts .....so out of the many problems i will have in the future ...this is my first one ...

the instruction is::find the derivative of the function :: y = 3x^3 + 2x^2 + x + 5

so using f'(x)= lim h-->0 -- f(x+h)-f(x)/h

i come to:: [ 3(x+h)^3 + 2(x+h)^2 + (x+h) + 5] - [3x^3 + 2x^2 + x + 5] / h

now this is where i was stuck ....i dont want to expand all that stuff cuz that would take not forever ..but a significant amt of time ...so i wanted to ask ...are there any shortcuts to come to the expanded form?

i solved this problem using the first 4 derivitives i learned y=c, y=cx, y=cx^n and another one ...but i wanted to know how to do this algebraically


thanks

[snipez90]

That is really not that bad of an expansion. For a binomial raised to the 4th or greater power, it wouldn't be bad if you knew the binomial theorem. But expansion of the third power or lower is easy.

Simply put, (x+y)^3 = x^3 + 3x^2y + 3xy^2 + y^3

But most people notice the pattern of polynomial differentiation after a few exercises. Try expanding and using the definition of the derivative and watch how all the terms cancel out. Then compare each term in the derivative with that of the original function and you should notice the pattern. Then look for a proof of it (it's quite easy if you know induction).

[chislam]

Learn the product rule for finding derivatives (specifically, that's the only rule you will need for this problem). The limit of the difference quotient is taught to you so that you know the definition of a derivative. Once, you have that nailed, there are plenty of "shortcuts" that you will be able to learn, such as the product rule, quotient rule, chain rule, etc.

[NoMoreExams]

Also derivative is "linear" in the sense that if D is your differential operator then D(f + g) = D(f) + D(g) and D(alpha*f) = alpha*D(f) where alpha is constant and f, g are your functions. So for example derivative of 3x^3 + 2x^2 is 3*(derivative of x^3) + 2*(derivative of x^2). Hope this helps. As a good exercise find the derivative of x^n (use binomial theorem to expand it out).