Find the Derivative Using Product Rule - d/dx (5x)/(2x^2)

Problem

d()/d(x)(5*x)/(2*x2)

Solution

1. Identify the functions for the product rule by rewriting the expression as a product of two terms.

ƒ(x)=5*x

g(x)=(2*x2)(−1)=1/2*x(−2)

2. Differentiate each function individually.

d(ƒ(x))/d(x)=5

d(g(x))/d(x)=−x(−3)

3. Apply the product rule formula, which is d()/d(x)*ƒ(x)*g(x)=ƒ(x)′*g(x)+ƒ(x)*g(x)′

d()/d(x)*(5*x)*(1/2*x(−2))=(5)*(1/2*x(−2))+(5*x)*(−x(−3))

4. Simplify the resulting terms.

5/(2*x2)−(5*x)/(x3)

5/(2*x2)−5/(x2)

5. Combine the fractions using a common denominator.

5/(2*x2)−10/(2*x2)

−5/(2*x2)

Final Answer

d()/d(x)(5*x)/(2*x2)=−5/(2*x2)

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