Differentiate the function.
y = (Square root x)(x − 10)
use the product rule on
y = (x^(1/2))(x-10)
dy/dx = x^(1/2) (1) + (1/2)x^(-1/2) (x-10)
= (1/2) x^(-1/2) ( 2x + x-10)
= (3x-10)/(2√x)
Thank you so much! I didn't think to use the product rule.
To differentiate the function y = √x (x - 10), we can apply the product rule and the chain rule.
The product rule states that if we have two functions u(x) and v(x), their derivative w(x) = u(x)v'(x) + u'(x)v(x), where u'(x) and v'(x) represent the derivatives of u(x) and v(x) with respect to x, respectively.
Let's first find the derivative of each part separately:
1. The derivative of √x is 1/2√x using the power rule.
2. The derivative of (x - 10) is simply 1 using the constant rule.
Now, let's use these derivatives to apply the product rule:
y' = (√x)(1) + (1/2√x)(x - 10)
Simplifying this expression, we get:
y' = √x + (x - 10)/(2√x)
Thus, the derivative of the function y = √x (x - 10) is y' = √x + (x - 10)/(2√x).