Find the Derivative - d/dx x/( square root of 7-3x)

Problem

d()/d(x)x/√(,7−3*x)

Solution

1. Identify the rule needed for differentiation, which is the quotient rule: d()/d(x)u/v=(vd(u)/d(x)−ud(v)/d(x))/(v2)

2. Assign the variables where u=x and v=√(,7−3*x)=(7−3*x)(1/2)

3. Differentiate u to find d(u)/d(x)=1

4. Differentiate v using the chain rule to find d(v)/d(x)=1/2*(7−3*x)(−1/2)⋅(−3)=−3/(2√(,7−3*x))

5. Substitute these components into the quotient rule formula:

d()/d(x)x/√(,7−3*x)=(√(,7−3*x)*(1)−x*(−3/(2√(,7−3*x))))/((√(,7−3*x))2)

6. Simplify the numerator by finding a common denominator of 2√(,7−3*x)

d()/d(x)x/√(,7−3*x)=(2*(7−3*x)+3*x)/(2√(,7−3*x))/(7−3*x)

7. Combine the terms in the numerator:

d()/d(x)x/√(,7−3*x)=(14−6*x+3*x)/(2*(7−3*x)(3/2))

8. Finalize the expression by simplifying the linear terms:

d()/d(x)x/√(,7−3*x)=(14−3*x)/(2*(7−3*x)(3/2))

Final Answer

d()/d(x)x/√(,7−3*x)=(14−3*x)/(2*(7−3*x)(3/2))

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