Find the Derivative - d/dx y = square root of 5-9x

Problem

d()/d(x)√(,5−9*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=5−9*x

2. Rewrite the radical expression using a fractional exponent to make it easier to differentiate.

√(,5−9*x)=(5−9*x)(1/2)

3. Apply the chain rule, which states that the derivative of ƒ*(g(x)) is ƒ′*(g(x))⋅g(x)′

d(5−9*x)/d(x)=1/2*(5−9*x)(−1/2)⋅d(5−9*x)/d(x)

4. Differentiate the inner function 5 - 9xw*i*t*h*r*e*s(p)*e*c*t*t*o$.

d(5−9*x)/d(x)=−9

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

1/2*(5−9*x)(−1/2)⋅(−9)

6. Simplify the expression by multiplying the constants and moving the negative exponent to the denominator as a square root.

−9/(2√(,5−9*x))

Final Answer

d(√(,5−9*x))/d(x)=−9/(2√(,5−9*x))

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