Solve d/dx12x+23/x

$$\frac { d } { d x } \frac { 12 x + 23 } { x }$$

Evaluate

$$-\frac{23}{x^{2}}$$

Steps Using Derivative Rule for Product

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.

$$\left(12x^{1}+23\right)\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{x})+\frac{1}{x}\frac{\mathrm{d}}{\mathrm{d}x}(12x^{1}+23)$$

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}.

$$\left(12x^{1}+23\right)\left(-1\right)x^{-1-1}+\frac{1}{x}\times 12x^{1-1}$$

Simplify.

$$\left(12x^{1}+23\right)\left(-1\right)x^{-2}+\frac{1}{x}\times 12x^{0}$$

Multiply 12x^{1}+23 times -x^{-2}.

$$12x^{1}\left(-1\right)x^{-2}+23\left(-1\right)x^{-2}+\frac{1}{x}\times 12x^{0}$$

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

$$-12x^{1-2}-23x^{-2}+12\times \frac{1}{x}$$

Simplify.

$$-12\times \frac{1}{x}-23x^{-2}+12\times \frac{1}{x}$$

Steps Using Derivative Rule for Quotient

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.

$$\frac{x^{1}\frac{\mathrm{d}}{\mathrm{d}x}(12x^{1}+23)-\left(12x^{1}+23\right)\frac{\mathrm{d}}{\mathrm{d}x}(x^{1})}{\left(x^{1}\right)^{2}}$$

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}.

$$\frac{x^{1}\times 12x^{1-1}-\left(12x^{1}+23\right)x^{1-1}}{\left(x^{1}\right)^{2}}$$

Do the arithmetic.

$$\frac{x^{1}\times 12x^{0}-\left(12x^{1}+23\right)x^{0}}{\left(x^{1}\right)^{2}}$$

Expand using distributive property.

$$\frac{x^{1}\times 12x^{0}-\left(12x^{1}x^{0}+23x^{0}\right)}{\left(x^{1}\right)^{2}}$$

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

$$\frac{12x^{1}-\left(12x^{1}+23x^{0}\right)}{\left(x^{1}\right)^{2}}$$

Remove unnecessary parentheses.

$$\frac{12x^{1}-12x^{1}-23x^{0}}{\left(x^{1}\right)^{2}}$$

Combine like terms.

$$\frac{\left(12-12\right)x^{1}-23x^{0}}{\left(x^{1}\right)^{2}}$$

Subtract 12 from 12.

$$-\frac{23x^{0}}{\left(x^{1}\right)^{2}}$$

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

$$-\frac{23x^{0}}{1^{2}x^{2}}$$

Raise 1 to the power 2.

$$-\frac{23x^{0}}{x^{2}}$$

Multiply 1 times 2.

$$\frac{-23x^{0}}{x^{2}}$$

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

$$\left(-\frac{23}{1}\right)x^{-2}$$

Do the arithmetic.

$$-23x^{-2}$$

Differentiate w.r.t. x

$$\frac{46}{x^{3}}$$