How are you left with 2x?


f(x+h)-f(x)
_________
h

2(x+h)-2(x)
________
h

2x+2h-2x
________
h

simplifies to.... just 2....

you messed up on your input function. Check Azrads post to see the correct function input and steps to take afterwards

There are lots of problems with what you wrote. But the first problem I would address is that f(x) is NOT 2x.... f(x) is 1/(2x)

As for #2:
if your teacher has not taught you how to find a derivative, then a=2 in the problem leads me to believe that he/she wants you to use the other limit definition formula


lim as x-->a of [f(x) - f(a)] / (x - a)

after plugging and chugging, you should get your slope (which is 20)

now that you have the slope, you need to find your y-value when x=2 (that means plug 2 into your original function and whatever y-value comes out).

now you should have your slope, and coordinates for a point on the line, so you have everything you need for the Point-Slope Formula:

y - y_1 = m(x - x_1)

then plug and chug again

How on earth do you get 20 for the first one, a is obviously two for f(a) you plug in two for the equation given and solve where do you get the x value?

Ok, so the problem says to use the rules of differentiation, not the limit definition, in order to find the tangent line to the function at a point, a.

f(x) = 3x^(3) - 4x^(2) + 2
a = 2

So, using the power rule (a rule of differentiation), you get
f'(x) = 9x^(2) - 8x

When you plug 2 into f'(x) (in order to find the slope of the original equation at x=2), you get
f'(x) = 9*2^(2) - 8*2 = 9*4-16 = 20.
So the slope of the tangent line is 20, therefore your line is y=20*x + b, and b is still unknown.

If you plug 2 into the function, you get
3*2^(3) - 4*2^(2) + 2 = 3*8 - 4*4 + 2 = 10
So the point on the curve is (x,y) = (2,10)

Plug these values into the tangent line equation.
10 = 20*2 + b and solve for b.
b = -30

So your equation is y = 20x - 30

What exactly does the power rule do?

where does the 2 end up? Im assuming the power rule you multiple the exponent by the constant?

here's an image of the scratchwork for #2



Now i see that the instructor doesn't want to use the limit definition, so using the Power Rule is your other option to find the derivative