Evaluate
\frac{5x+12}{x\left(x+6\right)}
Differentiate w.r.t. x
-\frac{5x^{2}+24x+72}{\left(x\left(x+6\right)\right)^{2}}
Graph
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\frac{3x}{x\left(x+6\right)}+\frac{2\left(x+6\right)}{x\left(x+6\right)}
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+6 and x is x\left(x+6\right). Multiply \frac{3}{x+6} times \frac{x}{x}. Multiply \frac{2}{x} times \frac{x+6}{x+6}.
\frac{3x+2\left(x+6\right)}{x\left(x+6\right)}
Since \frac{3x}{x\left(x+6\right)} and \frac{2\left(x+6\right)}{x\left(x+6\right)} have the same denominator, add them by adding their numerators.
\frac{3x+2x+12}{x\left(x+6\right)}
Do the multiplications in 3x+2\left(x+6\right).
\frac{5x+12}{x\left(x+6\right)}
Combine like terms in 3x+2x+12.
\frac{5x+12}{x^{2}+6x}
Expand x\left(x+6\right).
\frac{\mathrm{d}}{\mathrm{d}x}(\frac{3x}{x\left(x+6\right)}+\frac{2\left(x+6\right)}{x\left(x+6\right)})
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+6 and x is x\left(x+6\right). Multiply \frac{3}{x+6} times \frac{x}{x}. Multiply \frac{2}{x} times \frac{x+6}{x+6}.
\frac{\mathrm{d}}{\mathrm{d}x}(\frac{3x+2\left(x+6\right)}{x\left(x+6\right)})
Since \frac{3x}{x\left(x+6\right)} and \frac{2\left(x+6\right)}{x\left(x+6\right)} have the same denominator, add them by adding their numerators.
\frac{\mathrm{d}}{\mathrm{d}x}(\frac{3x+2x+12}{x\left(x+6\right)})
Do the multiplications in 3x+2\left(x+6\right).
\frac{\mathrm{d}}{\mathrm{d}x}(\frac{5x+12}{x\left(x+6\right)})
Combine like terms in 3x+2x+12.
\frac{\mathrm{d}}{\mathrm{d}x}(\frac{5x+12}{x^{2}+6x})
Use the distributive property to multiply x by x+6.
\frac{\left(x^{2}+6x^{1}\right)\frac{\mathrm{d}}{\mathrm{d}x}(5x^{1}+12)-\left(5x^{1}+12\right)\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}+6x^{1})}{\left(x^{2}+6x^{1}\right)^{2}}
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{\left(x^{2}+6x^{1}\right)\times 5x^{1-1}-\left(5x^{1}+12\right)\left(2x^{2-1}+6x^{1-1}\right)}{\left(x^{2}+6x^{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{\left(x^{2}+6x^{1}\right)\times 5x^{0}-\left(5x^{1}+12\right)\left(2x^{1}+6x^{0}\right)}{\left(x^{2}+6x^{1}\right)^{2}}
Simplify.
\frac{x^{2}\times 5x^{0}+6x^{1}\times 5x^{0}-\left(5x^{1}+12\right)\left(2x^{1}+6x^{0}\right)}{\left(x^{2}+6x^{1}\right)^{2}}
Multiply x^{2}+6x^{1} times 5x^{0}.
\frac{x^{2}\times 5x^{0}+6x^{1}\times 5x^{0}-\left(5x^{1}\times 2x^{1}+5x^{1}\times 6x^{0}+12\times 2x^{1}+12\times 6x^{0}\right)}{\left(x^{2}+6x^{1}\right)^{2}}
Multiply 5x^{1}+12 times 2x^{1}+6x^{0}.
\frac{5x^{2}+6\times 5x^{1}-\left(5\times 2x^{1+1}+5\times 6x^{1}+12\times 2x^{1}+12\times 6x^{0}\right)}{\left(x^{2}+6x^{1}\right)^{2}}
To multiply powers of the same base, add their exponents.
\frac{5x^{2}+30x^{1}-\left(10x^{2}+30x^{1}+24x^{1}+72x^{0}\right)}{\left(x^{2}+6x^{1}\right)^{2}}
Simplify.
\frac{-5x^{2}-24x^{1}-72x^{0}}{\left(x^{2}+6x^{1}\right)^{2}}
Combine like terms.
\frac{-5x^{2}-24x-72x^{0}}{\left(x^{2}+6x\right)^{2}}
For any term t, t^{1}=t.
\frac{-5x^{2}-24x-72}{\left(x^{2}+6x\right)^{2}}
For any term t except 0, t^{0}=1.
Examples
Quadratic equation
{ x } ^ { 2 } - 4 x - 5 = 0
Trigonometry
4 \sin \theta \cos \theta = 2 \sin \theta
Linear equation
y = 3x + 4
Arithmetic
699 * 533
Matrix
\left[ \begin{array} { l l } { 2 } & { 3 } \\ { 5 } & { 4 } \end{array} \right] \left[ \begin{array} { l l l } { 2 } & { 0 } & { 3 } \\ { -1 } & { 1 } & { 5 } \end{array} \right]
Simultaneous equation
\left. \begin{cases} { 8x+2y = 46 } \\ { 7x+3y = 47 } \end{cases} \right.
Differentiation
\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }
Integration
\int _ { 0 } ^ { 1 } x e ^ { - x ^ { 2 } } d x
Limits
\lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}+2 x-3}