Solve d/dxleft(55/25x-52right)

$$\frac{d}{d x } \left( \frac{ 55 }{ 25x } -52 \right)$$

Evaluate

$$-\frac{11}{5x^{2}}$$

Steps Using Derivative Rule for Quotient

To add or subtract expressions, expand them to make their denominators the same. Multiply 52 times \frac{25x}{25x}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{55}{25x}-\frac{52\times 25x}{25x})$$

Since \frac{55}{25x} and \frac{52\times 25x}{25x} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{55-52\times 25x}{25x})$$

Do the multiplications in 55-52\times 25x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{55-1300x}{25x})$$

Factor the expressions that are not already factored in \frac{55-1300x}{25x}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{5\left(-260x+11\right)}{25x})$$

Cancel out 5 in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{-260x+11}{5x})$$

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.

$$\frac{5x^{1}\frac{\mathrm{d}}{\mathrm{d}x}(-260x^{1}+11)-\left(-260x^{1}+11\right)\frac{\mathrm{d}}{\mathrm{d}x}(5x^{1})}{\left(5x^{1}\right)^{2}}$$

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}.

$$\frac{5x^{1}\left(-260\right)x^{1-1}-\left(-260x^{1}+11\right)\times 5x^{1-1}}{\left(5x^{1}\right)^{2}}$$

Do the arithmetic.

$$\frac{5x^{1}\left(-260\right)x^{0}-\left(-260x^{1}+11\right)\times 5x^{0}}{\left(5x^{1}\right)^{2}}$$

Expand using distributive property.

$$\frac{5x^{1}\left(-260\right)x^{0}-\left(-260x^{1}\times 5x^{0}+11\times 5x^{0}\right)}{\left(5x^{1}\right)^{2}}$$

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

$$\frac{5\left(-260\right)x^{1}-\left(-260\times 5x^{1}+11\times 5x^{0}\right)}{\left(5x^{1}\right)^{2}}$$

Do the arithmetic.

$$\frac{-1300x^{1}-\left(-1300x^{1}+55x^{0}\right)}{\left(5x^{1}\right)^{2}}$$

Remove unnecessary parentheses.

$$\frac{-1300x^{1}-\left(-1300x^{1}\right)-55x^{0}}{\left(5x^{1}\right)^{2}}$$

Combine like terms.

$$\frac{\left(-1300-\left(-1300\right)\right)x^{1}-55x^{0}}{\left(5x^{1}\right)^{2}}$$

Subtract -1300 from -1300.

$$-\frac{55x^{0}}{\left(5x^{1}\right)^{2}}$$

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

$$-\frac{55x^{0}}{5^{2}x^{2}}$$

Raise 5 to the power 2.

$$-\frac{55x^{0}}{25x^{2}}$$

Multiply 1 times 2.

$$\frac{-55x^{0}}{25x^{2}}$$

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

$$\left(-\frac{55}{25}\right)x^{-2}$$

Do the arithmetic.

$$-\frac{11}{5}x^{-2}$$

Differentiate w.r.t. x

$$\frac{22}{5x^{3}}$$