Find the Derivative - d/dx square root of x(x-10)

Problem

d(√(,x*(x−10)))/d(x)

Solution

1. Rewrite the expression using exponent notation and expand the product inside the square root.

√(,x*(x−10))=(x2−10*x)(1/2)

2. Apply the chain rule, which states that the derivative of un is n*u(n−1)⋅d(u)/d(x)

d(x2−10*x)/d(x)=1/2*(x2−10*x)(−1/2)⋅d(x2−10*x)/d(x)

3. Differentiate the inner function x2−10*x using the power rule.

d(x2−10*x)/d(x)=2*x−10

4. Substitute the inner derivative back into the chain rule expression.

1/2*(x2−10*x)(−1/2)⋅(2*x−10)

5. Simplify the expression by factoring out a 2 from the numerator and canceling it with the fraction.

(2*(x−5))/(2√(,x2−10*x))

6. Finalize the expression by canceling the common factor of 2.

(x−5)/√(,x2−10*x)

Final Answer

d(√(,x*(x−10)))/d(x)=(x−5)/√(,x2−10*x)

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