Solve [(x^{-2})^2+x^frac{11{2}}*x^-3]:x^-3

$$[ ( x ^ { - 2 } ) ^ { 2 } + x ^ { \frac { 11 } { 2 } } \cdot x ^ { - 3 } ] : x ^ { - 3 }$$

Evaluate

$$x^{\frac{11}{2}}+\frac{1}{x}$$

Short Solution Steps

To raise a power to another power, multiply the exponents. Multiply -2 and 2 to get -4.

$$\frac{x^{-4}+x^{\frac{11}{2}}x^{-3}}{x^{-3}}$$

To multiply powers of the same base, add their exponents. Add \frac{11}{2} and -3 to get \frac{5}{2}.

$$\frac{x^{-4}+x^{\frac{5}{2}}}{x^{-3}}$$

Factor the expressions that are not already factored.

$$\frac{x^{-4}\left(x^{\frac{13}{2}}+1\right)}{x^{-3}}$$

To divide powers of the same base, subtract the numerator's exponent from the denominator's exponent.

$$\frac{x^{\frac{13}{2}}+1}{x^{1}}$$

Expand the expression.

$$\frac{x^{\frac{13}{2}}+1}{x}$$

Differentiate w.r.t. x

$$\frac{11x^{\frac{9}{2}}}{2}-\frac{1}{x^{2}}$$

Steps Using Derivative Rule for Product

To raise a power to another power, multiply the exponents. Multiply -2 and 2 to get -4.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{-4}+x^{\frac{11}{2}}x^{-3}}{x^{-3}})$$

To multiply powers of the same base, add their exponents. Add \frac{11}{2} and -3 to get \frac{5}{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{-4}+x^{\frac{5}{2}}}{x^{-3}})$$

Factor the expressions that are not already factored in \frac{x^{-4}+x^{\frac{5}{2}}}{x^{-3}}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{-4}\left(x^{\frac{13}{2}}+1\right)}{x^{-3}})$$

To divide powers of the same base, subtract the numerator's exponent from the denominator's exponent.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{\frac{13}{2}}+1}{x^{1}})$$

Calculate x to the power of 1 and get x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{\frac{13}{2}}+1}{x})$$

For any two differentiable functions, the derivative of the product of two functions is the first function times the derivative of the second plus the second function times the derivative of the first.

$$\left(x^{\frac{13}{2}}+1\right)\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{x})+\frac{1}{x}\frac{\mathrm{d}}{\mathrm{d}x}(x^{\frac{13}{2}}+1)$$

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$\left(x^{\frac{13}{2}}+1\right)\left(-1\right)x^{-1-1}+\frac{1}{x}\times \frac{13}{2}x^{\frac{13}{2}-1}$$

Simplify.

$$\left(x^{\frac{13}{2}}+1\right)\left(-1\right)x^{-2}+\frac{1}{x}\times \frac{13}{2}x^{\frac{11}{2}}$$

Multiply x^{\frac{13}{2}}+1 times -x^{-2}.

$$x^{\frac{13}{2}}\left(-1\right)x^{-2}-x^{-2}+\frac{1}{x}\times \frac{13}{2}x^{\frac{11}{2}}$$

To multiply powers of the same base, add their exponents.

$$-x^{\frac{13}{2}-2}-x^{-2}+\frac{13}{2}x^{-1+\frac{11}{2}}$$

Simplify.

$$-x^{\frac{9}{2}}-x^{-2}+\frac{13}{2}x^{\frac{9}{2}}$$