9√x/x
a) rewrite
b)differentiate
c)simplify
To rewrite the expression 9√(x)/x, we can start by factoring out the square root from the numerator:
a) Rewrite:
= 9 * √(x)/x
To differentiate the expression, we will need to use the quotient rule. The quotient rule states that if we have a function f(x) = g(x)/h(x), then its derivative is given by:
f'(x) = (g'(x)*h(x) - g(x)*h'(x))/(h(x))^2
For our expression 9√(x)/x, we have:
g(x) = 9√(x)
h(x) = x
To find the derivatives g'(x) and h'(x), we need to apply the chain rule for differentiating the square root:
g'(x) = d/dx (9√(x))
= 9 * (1/2) * (x)^(-1/2)
= 9/(2√(x))
h'(x) = d/dx (x)
= 1
Now we can use the quotient rule to find f'(x):
f'(x) = (g'(x)*h(x) - g(x)*h'(x))/(h(x))^2
= (9/(2√(x)))(x) - (9√(x))(1)/x^2
= (9x)/(2√(x)) - 9√(x)/x^2
Now let's simplify the expression:
c) Simplify:
To simplify the expression 9x/(2√(x)) - 9√(x)/x^2, we can combine the terms by finding a common denominator:
= (9x^2 - 18√(x))/(2x^2√(x))
So, the simplified expression is (9x^2 - 18√(x))/(2x^2√(x)).