Find the Derivative - d/dx 1/( square root of x^17)

Problem

d()/d(x)1/√(,x17)

Solution

1. Rewrite the expression using exponents by noting that the square root is a power of 1/2

1/√(,x17)=1/((x17)(1/2))

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

1/((x17)(1/2))=1/(x(17/2))

3. Apply the negative exponent rule 1/(xn)=x(−n) to prepare the expression for differentiation.

x(−17/2)

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

d(x(−17/2))/d(x)=−17/2*x(−17/2−1)

5. Subtract the exponents by finding a common denominator.

−17/2−2/2=−19/2

6. Simplify the final expression by moving the variable back to the denominator.

−17/2*x(−19/2)=−17/(2*x(19/2))

Final Answer

d()/d(x)1/√(,x17)=−17/(2*x(19/2))

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