1. f(x)=√5x+6, find f'(x)
2. f(x)=√4x^2+3x+4, find f'(x).
i got stuck.
f(x)=√5x+6=(5x+6)^1/2
f'(x)= 1/2(5x+6)^-3/2*(5).
Almost. How did you get from ^1/2 to ^-3/2? 1/2 - 1 = -1/2.
f = (5x+6)^(1/2)
use the power rule and chain rule:
f' = 1/2 * (5x+6)^(-1/2) * 5 = 5/(2√(5x+6))
f = √(4x^2+3x+4)^(1/2)
f' = 1/2 * (4x^2+3x+4)^-1/2) * (8x+3) = (8x+3)/(2√(4x^2+3x+4))
oops.. i thought 1/2 to -3/2. now i understand. thanks
To find the derivative of a function like f(x)=√5x+6, you can use the chain rule of differentiation. The chain rule states that if we have a composite function, f(g(x)), then the derivative of f(g(x)) with respect to x is given by f'(g(x)) multiplied by the derivative of g(x) with respect to x.
Let's apply the chain rule to the first question:
1. f(x) = √5x+6
First, let's identify the inner function, g(x), which is 5x + 6. Then, let's find the derivative of g(x) with respect to x:
g'(x) = 5
Next, let's find the derivative of the outer function, f(g(x)), treating g(x) as the independent variable:
f'(g(x)) = 1/2(5x+6)^-1/2
Finally, we can apply the chain rule by multiplying the two derivatives together:
f'(x) = f'(g(x)) * g'(x) = 1/2(5x+6)^-1/2 * 5
Simplifying the expression gives us:
f'(x) = 5/2(5x+6)^-1/2
Now let's move on to the second question:
2. f(x) = √4x^2+3x+4
Similarly, we need to apply the chain rule to find the derivative. This time, the inner function, g(x), is 4x^2 + 3x + 4. Let's find its derivative:
g'(x) = 8x + 3
Now, let's find the derivative of the outer function, f(g(x)), treating g(x) as the independent variable:
f'(g(x)) = 1/2(4x^2+3x+4)^-1/2
Finally, applying the chain rule gives us:
f'(x) = f'(g(x)) * g'(x) = 1/2(4x^2+3x+4)^-1/2 * (8x + 3)
Simplifying this expression gives us the derivative of f(x).