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.