Using the formula lim h approaches 0 f(x+h)-f(x)/h, find the derivative of f(x)=2x^2+4x

just plug and chug:

f(x+h) = 2(x+h)^2+4(x+h)
= 2x^2+4xh+2h^2+4x+4h
now subtract f(x) = 2x^2+4x and you have
4xh+4h+2h^2

Now divide all that by h, and you have

4x+4+2h

lim as h->0 = 4x+4

To find the derivative of a function using the limit definition of the derivative, we can use the formula:

f'(x) = lim h→0 (f(x+h) - f(x)) / h

Let's apply this formula to the function f(x) = 2x^2 + 4x.

Step 1: Simplify the function
f(x) = 2x^2 + 4x

Step 2: Apply the limit definition of the derivative
f'(x) = lim h→0 [ f(x+h) - f(x) ] / h

Step 3: Substitute the function f(x) into the formula
f'(x) = lim h→0 [ (2(x+h)^2 + 4(x+h)) - (2x^2 + 4x) ] / h

Step 4: Expand and simplify the expression
f'(x) = lim h→0 [ (2(x^2 + 2xh + h^2) + 4x + 4h) - (2x^2 + 4x) ] / h
= lim h→0 [ 2x^2 + 4x + 4xh + 2h^2 + 4h - 2x^2 - 4x ] / h
= lim h→0 [ 4xh + 2h^2 + 4h ] / h

Step 5: Cancel out h from the expression
f'(x) = lim h→0 [4x + 2h + 4] = 4x + 4

Therefore, the derivative of f(x) = 2x^2 + 4x is f'(x) = 4x + 4.