Find the Derivative - d/dx 1/( square root of x^13)

Problem

d()/d(x)1/√(,x^13)

Solution

1. Rewrite the expression using exponent notation by converting the square root to a power of 1/2

1/√(,x^13)=1/((x^13)^(1/2))

2. Simplify the exponent using the power of a power rule, which states (a^m)^n=a^(m*n)

1/((x^13)^(1/2))=1/(x^(13/2))

3. Apply the negative exponent rule 1/(x^n)=x^(−n) to move the term to the numerator.

x^(−13/2)

4. Differentiate using the power rule d(x^n)/d(x)=n*x^(n−1)

d(x^(−13/2))/d(x)=−13/2*x^(−13/2−1)

5. Subtract the exponents by finding a common denominator.

−13/2−2/2=−15/2

6. Simplify the final expression by moving the negative exponent back to the denominator.

−13/(2*x^(15/2))

Final Answer

d()/d(x)1/√(,x^13)=−13/(2*x^(15/2))

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