Find the Derivative - d/d@VAR f(x) = natural log of square root of 3x

Problem

d(ln(√(,3*x)))/d(x)

Solution

1. Rewrite the expression using the properties of logarithms and exponents to simplify the differentiation process.

ƒ(x)=ln((3*x)(1/2))

2. Apply the power rule for logarithms, ln(ab)=b*ln(a) to move the exponent in front of the natural log.

ƒ(x)=1/2*ln(3*x)

3. Apply the product rule for logarithms, ln(a*b)=ln(a)+ln(b) to further decompose the expression.

ƒ(x)=1/2*(ln(3)+ln(x))

4. Differentiate the expression with respect to x noting that 1/2*ln(3) is a constant and its derivative is 0

d(ƒ(x))/d(x)=1/2*(0+d(ln(x))/d(x))

5. Substitute the derivative of ln(x) which is 1/x

d(ƒ(x))/d(x)=1/2⋅1/x

6. Simplify the resulting fraction.

d(ƒ(x))/d(x)=1/(2*x)

Final Answer

d(ln(√(,3*x)))/d(x)=1/(2*x)

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