Find the Derivative - d/dx f(x)=-3x square root of x+1

Problem

d()/d(x)*(−3*x√(,x+1))

Solution

1. Identify the function as a product of two terms, u=−3*x and v=√(,x+1) which requires the product rule d()/d(x)*u*v=ud(v)/d(x)+vd(u)/d(x)

2. Differentiate the first term u=−3*x with respect to x

(d(−)*3*x)/d(x)=−3

3. Differentiate the second term v=√(,x+1) using the chain rule, noting that √(,x+1)=(x+1)(1/2)

d(x+1)/d(x)=1/2*(x+1)(−1/2)

d(√(,x+1))/d(x)=1/(2√(,x+1))

4. Apply the product rule formula by substituting the derivatives found in the previous steps.

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

5. Simplify the expression by finding a common denominator to combine the terms.

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

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

d()/d(x)*(−3*x√(,x+1))=(−3*x−6*x−6)/(2√(,x+1))

d()/d(x)*(−3*x√(,x+1))=(−9*x−6)/(2√(,x+1))

Final Answer

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

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