Derivative of √ tan x is,
Answer: A. sin 2 x/ √ tan x B. cos 2 x/ 2 √ tan x C. sec 2 x D. None of these
if y = √tanx = (tanx)^(1/2)
then
dy/dx = (1/2)(tanx)^(-1/2)(sec^2 x)
or
sec^2x/(2√tanx)
I don't see that answer in your choices
You need to be more careful showing a ^ before exponents.
Hint: use the chain rule.
The correct answer is
(cosx)^-2/[2 sqrt(tanx)]
But that is none of the choices you have written down.
To find the derivative of √(tan x), we can use the chain rule of differentiation.
First, let's rewrite the expression as √(tan x) = (tan x)^(1/2).
Now, we can differentiate using the chain rule. The chain rule states that if we have a composite function f(g(x)), where f and g are differentiable functions, then the derivative of f(g(x)) is given by f'(g(x)) * g'(x).
Let's apply this to our function:
f(x) = (tan x)^(1/2)
Let's find f'(x):
f'(x) = (1/2) * (tan x)^(-1/2) * (sec^2 x)
We used the power rule to differentiate (tan x)^(1/2), which states that the derivative of x^n is nx^(n-1).
Next, let's simplify the expression by combining the factors:
f'(x) = (1/2) * (sec^2 x) / (tan x)^(1/2)
Now, let's rewrite (sec^2 x) as (1 + tan^2 x):
f'(x) = (1/2) * (1 + tan^2 x) / (tan x)^(1/2)
To simplify further, let's rewrite (tan x)^(1/2) as √(tan x):
f'(x) = (1/2) * (1 + tan^2 x) / √(tan x)
So, the derivative of √(tan x) is (1/2) * (1 + tan^2 x) / √(tan x).
Now let's compare the derivative we obtained with the options given:
A. sin 2 x / √ tan x
B. cos 2 x / 2 √ tan x
C. sec 2 x
D. None of these
Comparing the options with our derivative, we can see that none of the given options match our result. Therefore, the correct answer is D. None of these.