Find the derivative of f'(x) of f(x)=4sqrt(x) using limits
just use the difference quotient, and recall how to "rationalize" expressions involving roots:
√x - a = (√x-a)(√x+a)/(√x+a)
I did use the difference quotient but got a really weird answer. This was on a test I took earlier this morning and I know we get test corrections once the professor passes the test back
√(x+h)-√x
------------
h
(√(x+h)-√x)(√(x+h)+√x)
--------------------------
h(√(x+h)+√x)
(x+h)-x
-------------------
h(√(x+h)+√x)
h
---------------------
h(√(x+h)+√x)
The h's cancel, and you have
1/(√(x+h)+√x)
As h->0, that is 1/(√x+√x) = 1 / 2√x
To find the derivative of f(x) = 4√x using limits, we can use the definition of the derivative. The derivative of a function f(x) at a point x is defined as the limit of the difference quotient as h approaches 0:
f'(x) = lim(h->0) [f(x + h) - f(x)] / h
Let's apply this definition to find the derivative of f(x) = 4√x:
f(x) = 4√x
Step 1: Substitute x + h into the function f(x):
f(x + h) = 4√(x + h)
Step 2: Calculate the difference quotient:
[f(x + h) - f(x)] / h = [4√(x + h) - 4√x] / h
Step 3: Simplify the difference quotient:
[f(x + h) - f(x)] / h = [4√(x + h) - 4√x] / h
Step 4: Rationalize the numerator:
[f(x + h) - f(x)] / h = [4√(x + h) - 4√x] / h * (√(x + h) + √x) / (√(x + h) + √x)
Step 5: Simplify the numerator:
[f(x + h) - f(x)] / h = [4√(x + h) - 4√x] / h * (√(x + h) + √x) / (√(x + h) + √x)
= [4(x + h) - 4x] / h * (√(x + h) + √x) / (√(x + h) + √x)
= [4h] / h * (√(x + h) + √x) / (√(x + h) + √x)
= 4 * (√(x + h) + √x) / (√(x + h) + √x)
Step 6: Simplify further:
[f(x + h) - f(x)] / h = 4 * (√(x + h) + √x) / (√(x + h) + √x)
Step 7: Find the limit as h approaches 0:
f'(x) = lim(h->0) [f(x + h) - f(x)] / h
= lim(h->0) 4 * (√(x + h) + √x) / (√(x + h) + √x)
To evaluate this limit, notice that (√(x + h) + √x) is continuous and nonzero as h approaches 0. Therefore, we can substitute h = 0 into the expression:
f'(x) = 4 * (√(x + 0) + √x) / (√(x + 0) + √x)
= 4 * (√x + √x) / (√x + √x)
= 4 * 2√x / 2√x
= 4
Therefore, the derivative of f(x) = 4√x is f'(x) = 4. This means that the slope of the tangent line to the graph of f(x) is always 4.