Find the Derivative - d/dx (x^2+4x+3)/( square root of x)

Problem

d()/d(x)(x2+4*x+3)/√(,x)

Solution

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

(x2+4*x+3)/(x(1/2))=(x2)/(x(1/2))+(4*x)/(x(1/2))+3/(x(1/2))

2. Simplify each term using the laws of exponents by subtracting the power in the denominator from the power in the numerator.

x(3/2)+4*x(1/2)+3*x(−1/2)

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

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

4. Simplify the coefficients and rewrite the expression.

3/2*x(1/2)+2*x(−1/2)−3/2*x(−3/2)

5. Convert the terms back into radical form or a single fraction for the final result.

(3√(,x))/2+2/√(,x)−3/(2*x√(,x))

Final Answer

d()/d(x)(x2+4*x+3)/√(,x)=(3*x2+4*x−3)/(2*x√(,x))

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