Differentiate and simplify
(11x)(sqrt10x-5)
I think you use the Product rule but square roots confuse me with it being 1/2
To differentiate and simplify the expression (11x)(√(10x) - 5), we can use the Product Rule and the power rule for differentiation.
The Product Rule states that if you have a function u(x) multiplied by another function v(x), then the derivative of their product is given by (u(x) * v'(x)) + (v(x) * u'(x)).
Let's start by finding the derivatives of the functions in the expression:
The derivative of 11x with respect to x is simply 11.
To determine the derivative of √(10x) - 5, we first have to break it down into two parts: the square root function and the constant term.
The derivative of the square root function √(10x) with respect to x can be found using the Chain Rule. The Chain Rule states that if you have a composite function f(g(x)), then the derivative with respect to x is given by f'(g(x)) * g'(x).
In this case, f(x) = √x and g(x) = 10x. The derivative of f(x) = √x is (1/2)√x^(-1/2), and the derivative of g(x) = 10x is 10.
Therefore, the derivative of √(10x) with respect to x is (√x^(-1/2)) * 10 = 10/√(10x).
Now, let's simplify the expression using the Product Rule:
(11x)(√(10x) - 5) = (11x) * (√(10x)) - (11x) * 5
Differentiating the first term using the Product Rule, we get:
(11x) * (√(10x)) = (11x) * (10/√(10x)) = 110x/√(10x)
Differentiating the second term, we have:
(11x) * 5 = 55x
Hence, the final simplified expression is:
110x/√(10x) - 55x