Question

In Exercises, find the second derivative of the function. $$ f(x)=x \ln \sqrt{x}+2 x $$

Short answer

The second derivative of the function \( f(x)=x \ln \sqrt{x}+2x \) is \( f''(x)= 7 / (8*\sqrt{x^3}) \).

Step-by-step solution

Differentiate the function f(x)

Differentiate \( f(x)=x \ln \sqrt{x}+2x \) with respect to \( x \). This function is the sum of two terms, so it is differentiated term by term. 1) For \( x \ln \sqrt{x} \), apply the product rule for differentiation, which is \( (uv)' = u'v + uv' \) in this case \( u=x \) and \( v=\ln{\sqrt{x}} \). Differentiate \( x \) to get \( dx/dx = 1 \). Differentiate \( \ln{\sqrt{x}} \) using the chain rule, which is \( (f(g(x)))' = f'(g(x)) * g'(x) \), where \( f(u)=\ln{u} \) and \( g(x)=\sqrt{x} \). To differentiate \( \ln{u} \), we get \( du/u \). For \( \sqrt{x} \), differentiate it to get \( 1/2\sqrt{x} \).Combining these, the derivative of \( x \ln \sqrt{x} \) is \( 1*\ln{\sqrt{x}} + x*(1/(2*{\sqrt{x}}*{\sqrt{x}})) \), which simplifies to \( \ln{\sqrt{x}} + 1/(4\sqrt{x}) \).2) For \( 2x \), just apply the power rule. The derivative of \( x^n \), where \( n \) is a real number, is \( n*x^{(n-1)} \). In this case \( n=1 \), so the derivative of \( 2x \) is simply 2.The derivative of the function \( f(x) \) is the sum of these two derivatives: \( f'(x)=\ln{\sqrt{x}} + 1/(4\sqrt{x}) + 2 \).

Differentiate the derivative

The problem now asks for the second derivative of \( f(x) \), which is the derivative of the first derivative. Apply similar rules as in step 1. For the term \( \ln{\sqrt{x}} \), the derivative is \( 1/\sqrt{x} \).For the term \( 1/(4\sqrt{x}) \), apply the power rule to differentiate. Rewriting it as \( 1/4 * x^{-1/2} \), its derivative is \( -(1/2)*1/4*x^{-3/2} = -1/(8*x^{3/2}) \), or \( -1/(8\sqrt{x^3}) \). The derivative of 2 (a constant) is 0.Hence, the second derivative of \( f(x) \) is \( f''(x)= 1/\sqrt{x} - 1/(8\sqrt{x^3}) \).

Simplify the result

Combine the two terms from Step 2 under a common denominator to simplify the answer. Both terms have \( \sqrt{x} \) and \( \sqrt{x^3} \) in the denominator, so the least common denominator is \( 8*\sqrt{x^3} \).Therefore, the simplified form of the second derivative, \( f''(x) \), is \( (8 - 1) / (8*\sqrt{x^3}) = 7 / (8*\sqrt{x^3}) \).