Find the Derivative - d/dx y=x^2 square root of 10x-3

Problem

d()/d(x)*(x2√(,10*x−3))

Solution

1. Identify the function as a product of two terms, u=x2 and v=√(,10*x−3) which requires the product rule: d()/d(x)*(u*v)=ud(v)/d(x)+vd(u)/d(x)

2. Differentiate the first term u=x2 using the power rule.

d(x2)/d(x)=2*x

3. Differentiate the second term v=√(,10*x−3)=(10*x−3)(1/2) using the chain rule.

d(10*x−3)/d(x)=1/2*(10*x−3)(−1/2)⋅10

d(√(,10*x−3))/d(x)=5/√(,10*x−3)

4. Apply the product rule by combining the derivatives.

d(y)/d(x)=x2⋅5/√(,10*x−3)+√(,10*x−3)⋅2*x

5. Simplify the expression by finding a common denominator.

d(y)/d(x)=(5*x2)/√(,10*x−3)+(2*x*(10*x−3))/√(,10*x−3)

d(y)/d(x)=(5*x2+20*x2−6*x)/√(,10*x−3)

d(y)/d(x)=(25*x2−6*x)/√(,10*x−3)

Final Answer

d()/d(x)*(x2√(,10*x−3))=(25*x2−6*x)/√(,10*x−3)

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