Find (d)/(dx)(sqrt(49x^(-9)))

Question

Find  (d)/(dx)(sqrt(49x^(-9)))

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Identify Function: Identify the function to differentiate. The function given is sqrt(49x^(-9)).Rewrite Function: Rewrite the function in a form that is easier to differentiate. The square root of a number is the same as raising that number to the power of 1/2. So, sqrt(49x^(-9)) can be written as (49x^(-9))^(1/2).Apply Chain Rule: Apply the chain rule for differentiation. The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. In this case, the outer function is u^(1/2) and the inner function is 49x^(-9).Differentiate Outer Function: Differentiate the outer function with respect to the inner function. The derivative of u^(1/2) with respect to u is (1/2)u^(-1/2).Differentiate Inner Function: Differentiate the inner function with respect to x. The derivative of 49x^(-9) with respect to x is -9 * 49x^(-10).Combine Derivatives: Combine the derivatives using the chain rule. The combined derivative is (1/2)(49x^(-9))^(-1/2) * (-9 * 49x^(-10)).Simplify Expression: Simplify the expression. First, simplify (49x^(-9))^(-1/2) to (49^(-1/2)x^(9/2)). Then, multiply this by -9 * 49x^(-10).Further Simplify Expression: Further simplify the expression. Since 49^(-1/2) is the same as 1/sqrt(49) or 1/7, the expression becomes (1/7)x^(9/2) * (-9 * 49x^(-10)).Multiply Constants and Combine Terms: Multiply the constants and combine the x terms. The constants are (1/7) * (-9) * 49. The x terms are x^(9/2) * x^(-10), which simplifies to x^(-11/2).Complete Simplification: Complete the simplification. The constants (1/7) * (-9) * 49 simplify to -63. The final expression is -63x^(-11/2).

Find $\frac{d}{dx}\sqrt{49x^{-9}}$

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Find $\frac{d}{dx}\sqrt{49x^{-9}}$