Calculus>Derivative. Is there a technique to easily find the derivative?

Question

Our teacher in calculus said that there is a shorter technique to find the derivative of F, but he didn't teach it yet and he said that in our quiz we should use the long method. I should learn the short technique in order to check if my answer is correct because the long method is so complicated that if would occur any mistake it would make my answer wrong. So please do help me.

Can a scientific calculor solve derivative?
and if yes. How?If not just show the technique

for example find the derivative of F is F(x) = 3x^2 + 12

Answer 1

Yes there are shortcuts, but it is standard to learn the limit definition of a derivative first. You will begin learning the shortcuts right afterwards - starting with the power rule, product rule, quotient rule, and chain rule and moving onto other things.

And yes, you can do it on a calculator, but why would you want to? That takes the fun out of calculus!



Here are the two ways to find the derivative of your example:

Limit Definition:
F(x) = 3x^2 + 12
F(x + h) = 3(x + h)^2 + 12 = 3x^2 + 6xh + 3h^2 + 12

[F(x+h) - F(x)] / h
= [(3x^2 + 6xh + 3h^2 + 12) - (3x^2 + 12)] / h
= (6xh + 3h^2) / h
= 6x + 3h

limit as h-->0 of (6x + 3h) is 6x


The power rule:
d/dx[F(x)] = d/dx[3x^2 + 12]
F'(x) = (2*3)x^(2-1) + 0
= 6x

Answer 2

It depends what techniques you've already learned, and what you mean by the long way.

If the long way is to find (f(x+delta x)-f(x))/(delta x), then there is a short way, at least for most commonly occurring functions.

The short cut for your particular function is that the derivative of x^n is n x^(n-1)

There are definitely some computer programs out there, like mathematica and maple that will do derivatives. A lot of graphing calculators will compute the value of a derivative at a given value of x, but I don't know of any that will actually tell you the function in an algebraic form.

Answer 3

The calculator can evaluate the numerical value of derivitive at a given value of the independent variable. However, it doesn't do formulas.

The "short" method to find F'(x) is that f'(x)= nax^(n-1) if f(x)= ax^n. With this problem f'(x)= 6x. This is called the POWER RULE.
In the "long technique" you would substitute (x+del x) in the formula and subtract 3x^2-12. Then you would take the limit as delx approached zero.

Answer 4

f(x) = 3x^2 + 12

f'(x) = d/dx (3x^2 + 12)

derivative of a sum is the sum of the derivative

f'(x) = d/dx (3x^2) + d/dx (12)

memorize this rule:
d/dx u^n = n (u)^(n - 1) d/dx (u)

the derivative of a constant times a function is the constant times the derivative of the function

and, the derivative of a constant is 0

f'(x) = 3 d/dx (x^2)

f'(x) = 3 * 2x d/dx (x)

f'(x) = 6x