(d)/(dx)(sqrt(x^(5)))=

Rewrite Function: We are asked to find the derivative of the function sqrt(x^(5)) with respect to x. The function can be rewritten as (x^(5))^(1/2) to make differentiation easier.Differentiate Outer Function: Using the chain rule, we differentiate the outer function first, which is the square root function. The derivative of (u)^(1/2) with respect to u is (1/2)u^(-1/2).Differentiate Inner Function: Now we differentiate the inner function, which is x^(5). The derivative of x^(5) with respect to x is 5x^(4).Apply Chain Rule: Applying the chain rule, we multiply the derivative of the outer function by the derivative of the inner function. This gives us (1/2)(x^(5))^(-1/2) * 5x^(4).Combine Exponents: Simplify the expression by combining the exponents and coefficients. This results in (5/2)x^(4) * x^(-5/2).Add Exponents: When multiplying terms with the same base, we add the exponents. This gives us (5/2)x^(4 - 5/2) which simplifies to (5/2)x^(8/2 - 5/2).Simplify Exponent: Simplify the exponent by subtracting the fractions. This results in (5/2)x^(3/2).

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$\frac{d}{d x}\left(\sqrt{x^{5}}\right)=$

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Algebra 2

Simplify variable expressions using properties

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\frac{d}{d x}\left(\sqrt{x^{5}}\right)=

Rewrite Function: We are asked to find the derivative of the function $\sqrt{x^{5}}$ with respect to $x$ . The function can be rewritten as $(x^{5})^{\frac{1}{2}}$ to make differentiation easier.
Differentiate Outer Function: Using the chain rule, we differentiate the outer function first, which is the square root function. The derivative of $(u)^{\frac{1}{2}}$ with respect to $u$ is $\frac{1}{2}u^{-\frac{1}{2}}$ .
Differentiate Inner Function: Now we differentiate the inner function, which is $x^{5}$ . The derivative of $x^{5}$ with respect to $x$ is $5x^{4}$ .
Apply Chain Rule: Applying the chain rule, we multiply the derivative of the outer function by the derivative of the inner function. This gives us $(\frac{1}{2})(x^{5})^{(-\frac{1}{2})} \times 5x^{4}$ .
Combine Exponents: Simplify the expression by combining the exponents and coefficients. This results in $(\frac{5}{2})x^{4} \times x^{(-\frac{5}{2})}$ .
Add Exponents: When multiplying terms with the same base, we add the exponents. This gives us $(\frac{5}{2})x^{(4 - \frac{5}{2})}$ which simplifies to $(\frac{5}{2})x^{(\frac{8}{2} - \frac{5}{2})}$ .
Simplify Exponent: Simplify the exponent by subtracting the fractions. This results in $(\frac{5}{2})x^{(\frac{3}{2})}$ .