Find f '(a).
f(x)= sqrt(1 − 6x)
To find the derivative of f(x) with respect to x, we can use the power rule for differentiation. The power rule states that if we have a function of the form f(x) = x^n, then the derivative is given by f'(x) = nx^(n-1).
In this case, our function f(x) = √(1 - 6x) can be written as f(x) = (1 - 6x)^(1/2). To find f'(x), we need to apply the power rule.
Taking the derivative, we have:
f'(x) = (1/2)(1 - 6x)^(-1/2) * (-6)
Simplifying further, we get:
f'(x) = -3 / (1 - 6x)^(1/2)
To find f'(a), we substitute x with a in the expression we just derived:
f'(a) = -3 / (1 - 6a)^(1/2)
Therefore, f'(a) = -3 / √(1 - 6a).
To find f'(a), we need to take the derivative of the function f(x) with respect to x and then evaluate it at x = a.
The given function is f(x) = sqrt(1 - 6x).
To find the derivative, we can use the chain rule, which states that if we have a composition of functions, then the derivative is the derivative of the outer function multiplied by the derivative of the inner function.
Let's break down the problem step-by-step:
Step 1: Find the derivative of the outer function.
The derivative of the square root function is given by:
d/dx(sqrt(u)) = (1/2) * (1/sqrt(u)) * du/dx, where u is the inner function.
In this case, our inner function is 1 - 6x. Therefore, we have:
du/dx = d/dx(1 - 6x) = -6.
So, the derivative of the outer function is:
(1/2) * (1/sqrt(1 - 6x)) * (-6).
Step 2: Evaluate the derivative at x = a.
To find f'(a), we substitute x = a into the derivative expression:
f'(a) = (1/2) * (1/sqrt(1 - 6a)) * (-6).
Thus, f'(a) = -3/sqrt(1 - 6a).
Therefore, the derivative of f(x) = sqrt(1 - 6x) with respect to x is f'(x) = -3/sqrt(1 - 6x), and f'(a) = -3/sqrt(1 - 6a).