Solve d/dx(3x^2+9/x)

$$\frac { d } { d x } ( \frac { 3 x ^ { 2 } + 9 } { x } )$$

Evaluate

$$3-\frac{9}{x^{2}}$$

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.

$$\left(3x^{2}+9\right)\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{x})+\frac{1}{x}\frac{\mathrm{d}}{\mathrm{d}x}(3x^{2}+9)$$

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}.

$$\left(3x^{2}+9\right)\left(-1\right)x^{-1-1}+\frac{1}{x}\times 2\times 3x^{2-1}$$

Simplify.

$$\left(3x^{2}+9\right)\left(-1\right)x^{-2}+\frac{1}{x}\times 6x^{1}$$

Multiply 3x^{2}+9 times -x^{-2}.

$$3x^{2}\left(-1\right)x^{-2}+9\left(-1\right)x^{-2}+\frac{1}{x}\times 6x^{1}$$

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

$$-3x^{2-2}-9x^{-2}+6x^{-1+1}$$

Simplify.

$$-3x^{0}-9x^{-2}+6x^{0}$$

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

$$-3-9x^{-2}+6\times 1$$

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

$$-3-9x^{-2}+6$$

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.

$$\frac{x^{1}\frac{\mathrm{d}}{\mathrm{d}x}(3x^{2}+9)-\left(3x^{2}+9\right)\frac{\mathrm{d}}{\mathrm{d}x}(x^{1})}{\left(x^{1}\right)^{2}}$$

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}.

$$\frac{x^{1}\times 2\times 3x^{2-1}-\left(3x^{2}+9\right)x^{1-1}}{\left(x^{1}\right)^{2}}$$

Do the arithmetic.

$$\frac{x^{1}\times 6x^{1}-\left(3x^{2}+9\right)x^{0}}{\left(x^{1}\right)^{2}}$$

Expand using distributive property.

$$\frac{x^{1}\times 6x^{1}-\left(3x^{2}x^{0}+9x^{0}\right)}{\left(x^{1}\right)^{2}}$$

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

$$\frac{6x^{1+1}-\left(3x^{2}+9x^{0}\right)}{\left(x^{1}\right)^{2}}$$

Do the arithmetic.

$$\frac{6x^{2}-\left(3x^{2}+9x^{0}\right)}{\left(x^{1}\right)^{2}}$$

Remove unnecessary parentheses.

$$\frac{6x^{2}-3x^{2}-9x^{0}}{\left(x^{1}\right)^{2}}$$

Combine like terms.

$$\frac{\left(6-3\right)x^{2}-9x^{0}}{\left(x^{1}\right)^{2}}$$

Subtract 3 from 6.

$$\frac{3x^{2}-9x^{0}}{\left(x^{1}\right)^{2}}$$

Factor out 3.

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

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

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

Raise 1 to the power 2.

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

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

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

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

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

Differentiate w.r.t. x

$$\frac{18}{x^{3}}$$