Find du/dv u=v^2 square root of 8v-3

Problem

d()/d(v)*v2√(,8*v−3)

Solution

1. Identify the function as a product of two terms, ƒ(v)=v2 and g(v)=(8*v−3)(1/2) which requires the product rule: (d(u)*v)/d(v)=ud(v)/d(v)+vd(u)/d(v)

2. Differentiate the first part, v2 using the power rule to get 2*v

3. Differentiate the second part, (8*v−3)(1/2) using the chain rule. The derivative is 1/2*(8*v−3)(−1/2)⋅8 which simplifies to 4*(8*v−3)(−1/2)

4. Apply the product rule formula by combining the parts: v2⋅4*(8*v−3)(−1/2)+√(,8*v−3)⋅2*v

5. Simplify the expression by finding a common denominator, which is √(,8*v−3)

d(u)/d(v)=(4*v2)/√(,8*v−3)+2*v√(,8*v−3)

d(u)/d(v)=(4*v2+2*v*(8*v−3))/√(,8*v−3)

d(u)/d(v)=(4*v2+16*v2−6*v)/√(,8*v−3)

d(u)/d(v)=(20*v2−6*v)/√(,8*v−3)

Final Answer

d(u)/d(v)=(20*v2−6*v)/√(,8*v−3)

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