Find the Derivative - d/d@VAR f(x)=(3x^2-5)/(7x^2-2)

Problem

d()/d(x)(3*x2−5)/(7*x2−2)

Solution

1. Identify the function as a quotient of two differentiable functions, u=3*x2−5 and v=7*x2−2

2. Apply the quotient rule formula, which states that d()/d(x)u/v=(vd(u)/d(x)−ud(v)/d(x))/(v2)

3. Differentiate the numerator u to find d(u)/d(x)=6*x

4. Differentiate the denominator v to find d(v)/d(x)=14*x

5. Substitute these components into the quotient rule formula:

d(ƒ(x))/d(x)=((7*x2−2)*(6*x)−(3*x2−5)*(14*x))/((7*x2−2)2)

6. Distribute the terms in the numerator:

d(ƒ(x))/d(x)=(42*x3−12*x−(42*x3−70*x))/((7*x2−2)2)

7. Simplify the numerator by combining like terms:

d(ƒ(x))/d(x)=(42*x3−12*x−42*x3+70*x)/((7*x2−2)2)

d(ƒ(x))/d(x)=(58*x)/((7*x2−2)2)

Final Answer

d()/d(x)(3*x2−5)/(7*x2−2)=(58*x)/((7*x2−2)2)

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