Evaluate
\frac{1}{a+2}
Differentiate w.r.t. a
-\frac{1}{\left(a+2\right)^{2}}
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\frac{a}{\left(a-2\right)\left(a+2\right)}+\frac{2}{\left(a-2\right)\left(-a-2\right)}
Factor a^{2}-4. Factor 4-a^{2}.
\frac{a}{\left(a-2\right)\left(a+2\right)}+\frac{2\left(-1\right)}{\left(a-2\right)\left(a+2\right)}
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(a-2\right)\left(a+2\right) and \left(a-2\right)\left(-a-2\right) is \left(a-2\right)\left(a+2\right). Multiply \frac{2}{\left(a-2\right)\left(-a-2\right)} times \frac{-1}{-1}.
\frac{a+2\left(-1\right)}{\left(a-2\right)\left(a+2\right)}
Since \frac{a}{\left(a-2\right)\left(a+2\right)} and \frac{2\left(-1\right)}{\left(a-2\right)\left(a+2\right)} have the same denominator, add them by adding their numerators.
\frac{a-2}{\left(a-2\right)\left(a+2\right)}
Do the multiplications in a+2\left(-1\right).
\frac{1}{a+2}
Cancel out a-2 in both numerator and denominator.
\frac{\mathrm{d}}{\mathrm{d}a}(\frac{a}{\left(a-2\right)\left(a+2\right)}+\frac{2}{\left(a-2\right)\left(-a-2\right)})
Factor a^{2}-4. Factor 4-a^{2}.
\frac{\mathrm{d}}{\mathrm{d}a}(\frac{a}{\left(a-2\right)\left(a+2\right)}+\frac{2\left(-1\right)}{\left(a-2\right)\left(a+2\right)})
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(a-2\right)\left(a+2\right) and \left(a-2\right)\left(-a-2\right) is \left(a-2\right)\left(a+2\right). Multiply \frac{2}{\left(a-2\right)\left(-a-2\right)} times \frac{-1}{-1}.
\frac{\mathrm{d}}{\mathrm{d}a}(\frac{a+2\left(-1\right)}{\left(a-2\right)\left(a+2\right)})
Since \frac{a}{\left(a-2\right)\left(a+2\right)} and \frac{2\left(-1\right)}{\left(a-2\right)\left(a+2\right)} have the same denominator, add them by adding their numerators.
\frac{\mathrm{d}}{\mathrm{d}a}(\frac{a-2}{\left(a-2\right)\left(a+2\right)})
Do the multiplications in a+2\left(-1\right).
\frac{\mathrm{d}}{\mathrm{d}a}(\frac{1}{a+2})
Cancel out a-2 in both numerator and denominator.
-\left(a^{1}+2\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}a}(a^{1}+2)
If F is the composition of two differentiable functions f\left(u\right) and u=g\left(x\right), that is, if F\left(x\right)=f\left(g\left(x\right)\right), then the derivative of F is the derivative of f with respect to u times the derivative of g with respect to x, that is, \frac{\mathrm{d}}{\mathrm{d}x}(F)\left(x\right)=\frac{\mathrm{d}}{\mathrm{d}x}(f)\left(g\left(x\right)\right)\frac{\mathrm{d}}{\mathrm{d}x}(g)\left(x\right).
-\left(a^{1}+2\right)^{-2}a^{1-1}
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}.
-a^{0}\left(a^{1}+2\right)^{-2}
Simplify.
-a^{0}\left(a+2\right)^{-2}
For any term t, t^{1}=t.
-\left(a+2\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}