Find dy/dx y=(x^2+8x+3)/( square root of x)

Problem

d()/d(x)(x^2+8*x+3)/√(,x)

Solution

1. Rewrite the expression by expressing the square root as a power and splitting the fraction into individual terms.

y=(x^2)/(x^(1/2))+(8*x)/(x^(1/2))+3/(x^(1/2))

2. Simplify each term using the laws of exponents, specifically x^a/x^b=x^(a−b)

y=x^(3/2)+8*x^(1/2)+3*x^(−1/2)

3. Apply the power rule for differentiation, which states d(x^n)/d(x)=n*x^(n−1) to each term.

d(y)/d(x)=3/2*x^(1/2)+8*(1/2)*x^(−1/2)+3*(−1/2)*x^(−3/2)

4. Simplify the coefficients and rewrite the terms.

d(y)/d(x)=3/2*x^(1/2)+4*x^(−1/2)−3/2*x^(−3/2)

5. Factor out a common denominator to express the derivative as a single fraction.

d(y)/d(x)=(3*x^2+8*x−3)/(2*x^(3/2))

Final Answer

d(y)/d(x)=(3*x^2+8*x−3)/(2*x√(,x))

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