Differentiate the following function.
y = (5 x^2 + 7 x + 6)/sqrt(x)
use the quotient rule
h(x)=ƒ(x)/g(x)
h&prime(x)=[ƒ &prime (x)*g(x) - g&prime (x)*ƒ (x)]/g(x)2
To differentiate the given function y = (5x^2 + 7x + 6) / sqrt(x), we can apply the quotient rule of differentiation. The quotient rule states that if we have a function in the form of f(x) = g(x) / h(x), where g(x) and h(x) are both functions of x, then the derivative of f(x) can be found using the formula:
f'(x) = [g'(x) * h(x) - g(x) * h'(x)] / [h(x)]^2
Let's find the derivatives of the functions g(x) = 5x^2 + 7x + 6 and h(x) = sqrt(x).
1. Find g'(x):
To find the derivative of g(x), we can use the power rule for derivatives. The power rule states that if we have a function in the form f(x) = ax^n, where a is a constant and n is a real number, then the derivative of f(x) with respect to x is given by:
f'(x) = a * n * x^(n-1)
Using the power rule, we find that g'(x) = 10x + 7.
2. Find h'(x):
To find the derivative of h(x), which is sqrt(x), we can use the chain rule. The chain rule states that if we have a function in the form f(g(x)), then the derivative of f(g(x)) with respect to x is given by:
[f(g(x))]' = f'(g(x)) * g'(x)
In this case, f(u) = sqrt(u) and g(x) = x. Therefore, using the chain rule, we find that h'(x) = (1/2) * x^(-1/2) = 1 / (2 * sqrt(x)).
3. Plug the derivatives g'(x) and h'(x) into the quotient rule formula:
Now that we have the derivatives g'(x) = 10x + 7 and h'(x) = 1 / (2 * sqrt(x)), we can substitute them into the quotient rule formula:
f'(x) = [g'(x) * h(x) - g(x) * h'(x)] / [h(x)]^2
Plugging in the values, we get:
f'(x) = [(10x + 7) * sqrt(x) - (5x^2 + 7x + 6) * (1 / (2 * sqrt(x)))] / [sqrt(x)]^2
Simplifying further:
f'(x) = (10x * sqrt(x) + 7 * sqrt(x) - (5x^2 + 7x + 6) / (2 * sqrt(x))) / x
This is the derivative of the given function.
You could use the quotient rule, but why not try this approach
y = (5 x^2 + 7 x + 6)/sqrt(x)
= 5x^(3/2) + 7x^(1/2) + 6x^(-1/2)
now
dy/dx = (15/2)x^(1/2) + (7/2)x^(-1/2) - 3x^(-3/2)
= (1/2)x^(-3/2)[15x^2 + 7x - 6]
= (15x^2 + 7x - 6)/(2x√x)
check my algebra carefully.