Find the Derivative - d/dx y = square root of 6-9x

Problem

d()/d(x)√(,6−9*x)

Solution

1. Identify the function as a composition of functions, where the outer function is the square root u(1/2) and the inner function is u=6−9*x

2. Apply the power rule and the chain rule, which states that d()/d(x)*ƒ*(g(x))=ƒ′*(g(x))⋅g(x)′

3. Differentiate the outer function with respect to the inner function.

d()/d(u)*u(1/2)=1/2*u(−1/2)

4. Differentiate the inner function 6 - 9xw*i*t*h*r*e*s(p)*e*c*t*t*o$.

d()/d(x)*(6−9*x)=−9

5. Combine the results using the chain rule.

d(y)/d(x)=1/2*(6−9*x)(−1/2)⋅(−9)

6. Simplify the expression by moving the negative exponent to the denominator and multiplying the constants.

d(y)/d(x)=(−9)/(2√(,6−9*x))

Final Answer

d(√(,6−9*x))/d(x)=−9/(2√(,6−9*x))

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