find f ′(a) at the given number a. f(x) = sqrt(5x + 5), a = 4
f(x) = sqrt(5x+5)
f(a) = sqrt(5a+5)
f'(a) = 5/[2sqrt(5a+5)]
f'(4) = 5/[2sqrt(5(4)+5)]
f'(4) = 5/[2sqrt(20+5)]
f'(4) = 5/[2sqrt(25)]
f'(4) = 5/[2(5)]
f'(4) = 5/10
f'(4) = 1/2
f(x) = sqrt 5 * (x+1)^0.5
f'(x) = sqrt 5 * 0.5(x+1)^-0.5
f'(4) = 5^0.5 * 0.5 * 5^- 0.5 = 0.5
To find f' (a) at the given number a = 4 for the function f(x) = √(5x + 5), we need to find the derivative of f(x) with respect to x and then substitute x = 4.
Step 1: Find the derivative of f(x) with respect to x.
To find the derivative f'(x) of √(5x + 5), we can use the chain rule.
Let u = 5x + 5.
Now, rewrite f(x) = √u.
Applying the chain rule, we have: f'(x) = (1/2) * (u^(-1/2)) * u'.
Since u = 5x + 5, u' = d(5x + 5)/dx = 5.
Substituting the values, we have: f'(x) = (1/2) * (5x + 5)^(-1/2) * 5.
Step 2: Substitute x = 4 to find f' (a).
To find f' (a) at a = 4, substitute x = 4 in the expression we derived above for f'(x):
f'(4) = (1/2) * (5 * (4) + 5)^(-1/2) * 5.
Simplifying this expression, we get:
f'(4) = (1/2) * (20 + 5)^(-1/2) * 5.
f'(4) = (1/2) * (25)^(-1/2) * 5.
Now, calculating the values, we have:
f'(4) = (1/2) * (1/√25) * 5.
f'(4) = (1/2) * (1/5) * 5.
f'(4) = 1/2.
Therefore, f' (a) at a = 4 is 1/2.
To find the derivative of a function at a specific value, you can use the power rule and chain rule of differentiation. Here's how you can find f ′(a) for the given function f(x) = √(5x + 5) at a = 4:
1. Start with the given function: f(x) = √(5x + 5)
2. Apply the chain rule, which states that if you have a composition of functions, the derivative is the derivative of the outer function multiplied by the derivative of the inner function. In this case, the outer function is the square root function and the inner function is 5x + 5.
- The derivative of the outer function, the square root, is 1 / (2√(5x + 5)).
- The derivative of the inner function, 5x + 5, is simply 5.
3. Multiply the derivatives obtained from step 2: (1 / (2√(5x + 5))) * 5.
4. Substitute x = a = 4 into the derivative expression obtained from step 3: (1 / (2√(5*4 + 5))) * 5.
5. Simplify the expression to find f ′(a) at a = 4.
- Evaluate 5*4 + 5 = 20 + 5 = 25.
- The square root of 25 is 5, so the final expression becomes: (1 / (2 * 5)) * 5 = 1/2.
Therefore, f ′(4) = 1/2.