Find the Derivative - d/dx A(x)=120+12x+192/x

Problem

d()/d(x)*(120+12*x+192/x)

Solution

1. Rewrite the expression to make differentiation easier by expressing the fraction with a negative exponent.

A(x)=120+12*x+192*x(−1)

2. Apply the sum rule for derivatives, which allows for the differentiation of each term individually.

d(A(x))/d(x)=d(120)/d(x)+(d(12)*x)/d(x)+(d(192)*x(−1))/d(x)

3. Differentiate the constant term and the linear term using the constant rule and the power rule.

d(120)/d(x)=0

(d(12)*x)/d(x)=12

4. Apply the power rule to the term with the negative exponent, where d(xn)/d(x)=n*x(n−1)

(d(192)*x(−1))/d(x)=192*(−1)*x(−2)

(d(192)*x(−1))/d(x)=−192*x(−2)

5. Simplify the final expression by rewriting the negative exponent as a fraction.

d(A(x))/d(x)=12−192/(x2)

Final Answer

d()/d(x)*(120+12*x+192/x)=12−192/(x2)

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