if f(x) = square root of e^2x +1, then then f'(0) equals what?
f(x) = [e^2x + 1]^(1/2)
df/dx = [(1/2)/sqrt[e^2x + 1]* 2*e^2x = e^2x / sqrt[e^2x + 1]
= 1/sqrt2 at t = 0
To find the derivative of f(x) = sqrt(e^(2x) + 1), you can use the chain rule.
Step 1: Let's find the derivative of the expression inside the square root:
Let u = e^(2x) + 1, then du/dx = 2e^(2x)
Step 2: Apply the chain rule to find the derivative of the square root:
f'(x) = (1/2) * (e^(2x) + 1)^(-1/2) * 2e^(2x)
= e^(2x) / sqrt(e^(2x) + 1)
Step 3: Evaluate the derivative at x = 0:
f'(0) = e^(2(0)) / sqrt(e^(2(0)) + 1)
= e^0 / sqrt(e^0 + 1)
= 1 / sqrt(1 + 1)
= 1 / sqrt(2)
Therefore, f'(0) equals 1 / sqrt(2).
To find the derivative of the function f(x) = √(e^(2x) + 1), we can use the chain rule. The chain rule states that if we have a function of the form √(g(x)), then the derivative is given by [g'(x)] / [2√(g(x))].
Let's break it down step by step:
Step 1: Find the derivative of the function inside the square root: g'(x)
In this case, g(x) = e^(2x) + 1. To find g'(x), we differentiate e^(2x) + 1 separately.
The derivative of e^(2x) is obtained by applying the chain rule, which states that the derivative of e^(u(x)) is given by [u'(x)] * e^(u(x)). In this case, u(x) = 2x, so the derivative is 2 * e^(2x).
The derivative of 1 is 0 since it is a constant.
Therefore, g'(x) = 2 * e^(2x) + 0 = 2 * e^(2x).
Step 2: Find the derivative of the function f(x)
Apply the chain rule by substituting g(x) and g'(x) into the derivative formula: [g'(x)] / [2√(g(x))].
f'(x) = [2 * e^(2x)] / [2√(e^(2x) + 1)].
Step 3: Evaluate f'(0)
To find f'(0), substitute x = 0 into the derived expression:
f'(0) = [2 * e^(2 * 0)] / [2√(e^(2 * 0) + 1)]
= [2 * e^0] / [2√(e^0 + 1)]
= [2 * 1] / [2√(1 + 1)]
= 2 / [2 * √2]
= 2 / (2√2)
= 1 / √2
= √2 / 2.
Therefore, f'(0) equals √2 / 2.