Find the Derivative - d/dx (64x)/(x^2+64)

Problem

d()/d(x)(64*x)/(x2+64)

Solution

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

2. Assign the numerator and denominator functions: u=64*x and v=x2+64

3. Differentiate the individual components: d(u)/d(x)=64 and d(v)/d(x)=2*x

4. Substitute these values into the quotient rule formula:

((x2+64)*(64)−(64*x)*(2*x))/((x2+64)2)

5. Distribute the constant and multiply the terms in the numerator:

(64*x2+4096−128*x2)/((x2+64)2)

6. Combine like terms in the numerator to simplify the expression:

(4096−64*x2)/((x2+64)2)

7. Factor out the greatest common factor from the numerator:

(64*(64−x2))/((x2+64)2)

Final Answer

d()/d(x)(64*x)/(x2+64)=(64*(64−x2))/((x2+64)2)

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