Final answer:
To differentiate y = √(x)/4 + x, first express the square root as a power, apply the power rule, and then add the derivative of the linear term. The final derivative is y' = 1/(8√x) + 1.
Step-by-step explanation:
The student has asked to differentiate the function y = √(x)/4 + x. Assuming that the function is meant to be y = √(x)/4 + x as written (and not, for example, y = (√(x) + x)/4 due to typing error), we will find the derivative y' by applying the rules of differentiation to each term separately.
Steps for Differentiation
Identify the individual terms of the function, which are √(x)/4 and x.
For the first term, √(x)/4, rewrite the square root as a power, resulting in (x^0.5)/4.
Apply the power rule to differentiate x^0.5, which gives us 0.5x^(-0.5).
For the second term, the derivative of x with respect to x is simply 1.
Combine the differentiated terms and simplify if necessary, giving us the final derivative y' = (0.5x^(-0.5))/4 + 1 which can be also written as y' = 1/(8√x) + 1.
Therefore, the derivative of the function y = √(x)/4 + x with respect to x is y' = 1/(8√x) + 1.