Differentiate w.r.t. t
\frac{12\left(1-3t^{2}\right)}{\left(3t^{2}+1\right)^{2}}
Evaluate
\frac{12t}{3t^{2}+1}
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\frac{\left(3t^{2}+1\right)\frac{\mathrm{d}}{\mathrm{d}t}(12t^{1})-12t^{1}\frac{\mathrm{d}}{\mathrm{d}t}(3t^{2}+1)}{\left(3t^{2}+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(3t^{2}+1\right)\times 12t^{1-1}-12t^{1}\times 2\times 3t^{2-1}}{\left(3t^{2}+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(3t^{2}+1\right)\times 12t^{0}-12t^{1}\times 6t^{1}}{\left(3t^{2}+1\right)^{2}}
Do the arithmetic.
\frac{3t^{2}\times 12t^{0}+12t^{0}-12t^{1}\times 6t^{1}}{\left(3t^{2}+1\right)^{2}}
Expand using distributive property.
\frac{3\times 12t^{2}+12t^{0}-12\times 6t^{1+1}}{\left(3t^{2}+1\right)^{2}}
To multiply powers of the same base, add their exponents.
\frac{36t^{2}+12t^{0}-72t^{2}}{\left(3t^{2}+1\right)^{2}}
Do the arithmetic.
\frac{\left(36-72\right)t^{2}+12t^{0}}{\left(3t^{2}+1\right)^{2}}
Combine like terms.
\frac{-36t^{2}+12t^{0}}{\left(3t^{2}+1\right)^{2}}
Subtract 72 from 36.
\frac{12\left(-3t^{2}+t^{0}\right)}{\left(3t^{2}+1\right)^{2}}
Factor out 12.
\frac{12\left(-3t^{2}+1\right)}{\left(3t^{2}+1\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}