Find (d)/(dx)(sqrt(64x^(-3)))

Rewrite Function: We are given the function sqrt(64x^(-3)) and we need to find its derivative with respect to x. The function can be rewritten as (64x^(-3))^(1/2).Apply Chain Rule: Apply the chain rule for differentiation, which 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 64x^(-3).Derivative of Outer Function: Differentiate the outer function u^(1/2) with respect to u. The derivative is (1/2)u^(-1/2).Derivative of Inner Function: Differentiate the inner function 64x^(-3) with respect to x. The derivative is -3*64x^(-4) which simplifies to -192x^(-4).Combine Derivatives: Combine the derivatives of the outer and inner functions using the chain rule. Multiply (1/2)(64x^(-3))^(-1/2) by -192x^(-4).Simplify Expression: Simplify the expression. The x^(-3) term in the outer derivative and the x^(-4) term in the inner derivative combine to x^(-3-4) which is x^(-7). The constants 1/2 and -192 multiply to give -96. So the derivative is -96x^(-7).Final Derivative: The final simplified form of the derivative is -96/x^7 since x^(-7) is equivalent to 1/x^7.

Home

Math Problems

Precalculus

Evaluate rational exponents

Full solution

Q. Find

\frac{d}{dx}(\sqrt{64x^{-3}})

Rewrite Function: We are given the function $\sqrt{64x^{-3}}$ and we need to find its derivative with respect to $x$ . The function can be rewritten as $(64x^{-3})^{\frac{1}{2}}$ .
Apply Chain Rule: Apply the chain rule for differentiation, which 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 $64x^{-3}$ .
Derivative of Outer Function: Differentiate the outer function $u^{(1/2)}$ with respect to $u$ . The derivative is $(1/2)u^{(-1/2)}$ .
Derivative of Inner Function: Differentiate the inner function $64x^{-3}$ with respect to $x$ . The derivative is $-3\times64x^{-4}$ which simplifies to $-192x^{-4}$ .
Combine Derivatives: Combine the derivatives of the outer and inner functions using the chain rule. Multiply $(\frac{1}{2})(64x^{-3})^{-\frac{1}{2}}$ by $-192x^{-4}$ .
Simplify Expression: Simplify the expression. The $x^{-3}$ term in the outer derivative and the $x^{-4}$ term in the inner derivative combine to $x^{-3-4}$ which is $x^{-7}$ . The constants $\frac{1}{2}$ and $-192$ multiply to give $-96$ . So the derivative is $-96x^{-7}$ .
Final Derivative: The final simplified form of the derivative is $-96/x^7$ since $x^{-7}$ is equivalent to $1/x^7$ .