Find the Derivative - d/dx (x+7)/x+6/(3x+21)

Problem

d()/d(x)*((x+7)/x+6/(3*x+21))

Solution

1. Simplify the second term by factoring the denominator.

6/(3*x+21)=6/(3*(x+7))

2. Reduce the fraction by dividing the numerator and denominator by 3.

6/(3*(x+7))=2/(x+7)

3. Rewrite the first term by splitting the fraction.

(x+7)/x=x/x+7/x=1+7*x(−1)

4. Express the entire function in a form ready for differentiation.

ƒ(x)=1+7*x(−1)+2*(x+7)(−1)

5. Apply the power rule and the chain rule to differentiate each term.

d()/d(x)*1=0

d()/d(x)*7*x(−1)=−7*x(−2)

d()/d(x)*2*(x+7)(−1)=−2*(x+7)(−2)⋅1

6. Simplify the resulting expression back into fraction form.

−7/(x2)−2/((x+7)2)

Final Answer

d()/d(x)*((x+7)/x+6/(3*x+21))=−7/(x2)−2/((x+7)2)

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