If f(x)= sqrt(4sin(x)+2), then f'(0)=
Use the substitution
u=4sin(x)+2
f'(x)=df(x)/dx
=df(u)/du * du/dx
=d(sqrt(u))/du * d(4sin(x)+2)/dx
=1/(2sqrt(u)) * 4cos(x)
=2cos(x)/sqrt(4sin(x)+2))
Well the Sqrt(4sin(x)+2) is the same thing as saying (4sin(x)+2)^(1/2)....sooo take the derivative using the chain rule. So first you would take the derivative of the outside by bringing the (1/2) and subtracting one from the exponent. Then take the derivative of the inside. 4 is a constant so it remains the same since it being multiplied. Sin(x)has the derivative cos(x) and since the two is a constant its derivative is zero. Your answer should look like this:
(4cos(x))/(2sqrt(4sin(x)+2)
radical 2
rad(4sinx +2) or (4sinx +2)^1/2
chain rule
1/2(4sinx +2)^-(1/2) * 4cosx <----- derivative of sinx is cosx
substitue x for 0 and plug into calulator as follows:
1/2 (4sin(0) +2) ^-(1/2) *4cos(0)
final answer
f'(0) aprox. 1.414 or radical(2), squar root(2) same thing
Well, if we want to find f'(0), we need to take the derivative of f(x) with respect to x and then evaluate it at x=0.
Let's go step by step. The derivative of f(x) is given by:
f'(x) = (1/2) * (4sin(x)+2)^(-1/2) * (4cos(x))
Now, let's plug in x=0:
f'(0) = (1/2) * (4sin(0)+2)^(-1/2) * (4cos(0))
Simplifying that a bit:
f'(0) = (1/2) * (2)^(-1/2) * (4 * 1)
And rearranging:
f'(0) = (1/2) * (1/sqrt(2)) * 4
Doing some more math:
f'(0) = (1/2) * (2) * 4
Fortunately, math is not my strong suit and I've gone around in circles. So, let's analyze this situation from a clown perspective - f'(0) might just be some sort of secret code that actually stands for "funny lady with gingerbread cookies." Now doesn't that sound more exciting than calculating derivatives?
To find the derivative of a function at a specific point, you can use the derivative rules and techniques.
Let's find the derivative of f(x) = √(4sin(x) + 2) using the chain rule.
1. Begin by applying the chain rule. The chain rule states that the derivative of an outer function (in this case, the square root function) multiplied by the derivative of the inner function (in this case, 4sin(x) + 2) will give you the derivative of the composite function.
2. The derivative of the square root function (√u) is (1/2√u).
3. Differentiate the inner function, 4sin(x) + 2, to find its derivative. The derivative of sin(x) is cos(x).
4. Multiply the derivative of the outer function by the derivative of the inner function:
f'(x) = (1/2√(4sin(x) + 2)) * (4cos(x)).
Now, let's find f'(0) by substituting x = 0 into the derivative:
f'(0) = (1/2√(4sin(0) + 2)) * (4cos(0)).
Since sin(0) = 0 and cos(0) = 1, the equation simplifies to:
f'(0) = (1/2√2) * 4 * 1.
f'(0) = (1/√2) * 4.
To simplify further, we rationalize the denominator (√2) by multiplying the numerator and denominator by √2:
f'(0) = (4√2)/2.
Finally, we simplify:
f'(0) = 2√2.
Therefore, f'(0) = 2√2.