Using the formula lim h approaches 0 f(x+h)-f(x)/h, find the derivative of f(x)=2x^2+4x
It's 4x+4
Sure, I'll derive the equation for you! But first, let me put on my thinking wig...
Alright, let's get to clowning around with derivatives! We'll use the formula you provided, lim h approaches 0 f(x+h)-f(x)/h, to find the derivative of f(x) = 2x^2 + 4x.
Step 1: Let's plug in the values into the formula.
lim h approaches 0 [2(x+h)^2 + 4(x+h) - (2x^2 + 4x)] / h
Step 2: Time to expand and simplify!
lim h approaches 0 [2(x^2 + 2xh + h^2) + 4x + 4h - 2x^2 - 4x] / h
Step 3: Wave your magic wand and watch the simplification happen!
lim h approaches 0 (2x^2 + 4x + 2h^2 + 4h - 2x^2 - 4x) / h
Step 4: Let's get rid of the unnecessary terms!
lim h approaches 0 (2h^2 + 4h) / h
Step 5: Keep calm and divide!
lim h approaches 0 (2h(h + 2)) / h
Step 6: Canceling out the h's, and voila!
lim h approaches 0 2(h + 2)
Step 7: Finally, simplify to get the derivative!
2(0 + 2) = 4
So, ta-da! The derivative of f(x) = 2x^2 + 4x is 4. Remember, this is Clown Bot's take on derivatives, so feel free to laugh it off!
To find the derivative of the function f(x) = 2x^2 + 4x using the limit definition of derivative, we can follow these steps:
Step 1: Replace f(x) with the given function: f(x) = 2x^2 + 4x.
Step 2: Use the given formula lim h approaches 0 [f(x+h) - f(x)] / h.
Step 3: Substitute f(x + h) with the function for f(x + h): f(x + h) = 2(x + h)^2 + 4(x + h).
Step 4: Expand and simplify f(x + h):
f(x + h) = 2(x^2 + 2hx + h^2) + 4(x + h)
= 2x^2 + 4hx + 2h^2 + 4x + 4h
Step 5: Calculate f(x + h) - f(x):
f(x + h) - f(x) = (2x^2 + 4hx + 2h^2 + 4x + 4h) - (2x^2 + 4x)
= 2hx + 2h^2 + 4h
Step 6: Substitute back into the formula: [f(x + h) - f(x)] / h = (2hx + 2h^2 + 4h) / h.
Step 7: Simplify: (2hx + 2h^2 + 4h) / h = 2x + 2h + 4.
Step 8: Take the limit as h approaches 0: lim h approaches 0 (2x + 2h + 4) = 2x + 4.
Therefore, the derivative of f(x) = 2x^2 + 4x is equal to 2x + 4.
f(x+h) = 2(x+h)^2 + 4(x+h)
= 2x^2 + 4xh + 2h^2 + 4x + 4h
f ' (x) = lim (2x^2 + 4xh + 2h^2 + 4x + 4h - 2x^2 - 4x)/h , as h ---> 0
= lim (4xh + 2h^2 + 4h)/h
= lim h(4x + 2h + 4)/h
= lim 4x + 2h + 4 , as h --> 0
= 4x + 4