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

Problem

d()/d(x)1/√(,x13)

Solution

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

1/√(,x13)=1/((x13)(1/2))

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

1/((x13)(1/2))=1/(x(13/2))

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

x(−13/2)

4. Differentiate using the power rule d(xn)/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/√(,x13)=−13/(2*x(15/2))

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