Solve d/dxleft(2x^2+10000x/x^2right)

$$\frac{d}{d x } \left(2 { x }^{ 2 } + \frac{ 10000x }{ { x }^{ 2 } } \right)$$

Evaluate

$$4x-\frac{10000}{x^{2}}$$

Steps Using Derivative Rule for Product

Cancel out x in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(2x^{2}+\frac{10000}{x})$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 2x^{2} times \frac{x}{x}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2x^{2}x}{x}+\frac{10000}{x})$$

Since \frac{2x^{2}x}{x} and \frac{10000}{x} have the same denominator, add them by adding their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2x^{2}x+10000}{x})$$

Do the multiplications in 2x^{2}x+10000.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2x^{3}+10000}{x})$$

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.

$$\left(2x^{3}+10000\right)\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{x})+\frac{1}{x}\frac{\mathrm{d}}{\mathrm{d}x}(2x^{3}+10000)$$

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}.

$$\left(2x^{3}+10000\right)\left(-1\right)x^{-1-1}+\frac{1}{x}\times 3\times 2x^{3-1}$$

Simplify.

$$\left(2x^{3}+10000\right)\left(-1\right)x^{-2}+\frac{1}{x}\times 6x^{2}$$

Multiply 2x^{3}+10000 times -x^{-2}.

$$2x^{3}\left(-1\right)x^{-2}+10000\left(-1\right)x^{-2}+\frac{1}{x}\times 6x^{2}$$

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

$$-2x^{3-2}-10000x^{-2}+6x^{-1+2}$$

Simplify.

$$-2x^{1}-10000x^{-2}+6x^{1}$$

For any term t, t^{1}=t.

$$-2x-10000x^{-2}+6x$$

Steps Using Derivative Rule for Quotient

Cancel out x in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(2x^{2}+\frac{10000}{x})$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 2x^{2} times \frac{x}{x}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2x^{2}x}{x}+\frac{10000}{x})$$

Since \frac{2x^{2}x}{x} and \frac{10000}{x} have the same denominator, add them by adding their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2x^{2}x+10000}{x})$$

Do the multiplications in 2x^{2}x+10000.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2x^{3}+10000}{x})$$

For any two differentiable functions, the derivative of the quotient of two functions is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the denominator squared.

$$\frac{x^{1}\frac{\mathrm{d}}{\mathrm{d}x}(2x^{3}+10000)-\left(2x^{3}+10000\right)\frac{\mathrm{d}}{\mathrm{d}x}(x^{1})}{\left(x^{1}\right)^{2}}$$

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}.

$$\frac{x^{1}\times 3\times 2x^{3-1}-\left(2x^{3}+10000\right)x^{1-1}}{\left(x^{1}\right)^{2}}$$

Do the arithmetic.

$$\frac{x^{1}\times 6x^{2}-\left(2x^{3}+10000\right)x^{0}}{\left(x^{1}\right)^{2}}$$

Expand using distributive property.

$$\frac{x^{1}\times 6x^{2}-\left(2x^{3}x^{0}+10000x^{0}\right)}{\left(x^{1}\right)^{2}}$$

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

$$\frac{6x^{1+2}-\left(2x^{3}+10000x^{0}\right)}{\left(x^{1}\right)^{2}}$$

Do the arithmetic.

$$\frac{6x^{3}-\left(2x^{3}+10000x^{0}\right)}{\left(x^{1}\right)^{2}}$$

Remove unnecessary parentheses.

$$\frac{6x^{3}-2x^{3}-10000x^{0}}{\left(x^{1}\right)^{2}}$$

Combine like terms.

$$\frac{\left(6-2\right)x^{3}-10000x^{0}}{\left(x^{1}\right)^{2}}$$

Subtract 2 from 6.

$$\frac{4x^{3}-10000x^{0}}{\left(x^{1}\right)^{2}}$$

Factor out 4.

$$\frac{4\left(x^{3}-2500x^{0}\right)}{\left(x^{1}\right)^{2}}$$

To raise the product of two or more numbers to a power, raise each number to the power and take their product.

$$\frac{4\left(x^{3}-2500x^{0}\right)}{1^{2}x^{2}}$$

Raise 1 to the power 2.

$$\frac{4\left(x^{3}-2500x^{0}\right)}{x^{2}}$$

For any term t except 0, t^{0}=1.

$$\frac{4\left(x^{3}-2500\times 1\right)}{x^{2}}$$

For any term t, t\times 1=t and 1t=t.

$$\frac{4\left(x^{3}-2500\right)}{x^{2}}$$

Differentiate w.r.t. x

$$4+\frac{20000}{x^{3}}$$