Differentiate w.r.t. a
4a
Evaluate
2a^{2}
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2a^{3}\frac{\mathrm{d}}{\mathrm{d}a}(\frac{1}{a})+\frac{1}{a}\frac{\mathrm{d}}{\mathrm{d}a}(2a^{3})
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.
2a^{3}\left(-1\right)a^{-1-1}+\frac{1}{a}\times 3\times 2a^{3-1}
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}.
2a^{3}\left(-1\right)a^{-2}+\frac{1}{a}\times 6a^{2}
Simplify.
-2a^{3-2}+6a^{-1+2}
To multiply powers of the same base, add their exponents.
-2a^{1}+6a^{1}
Simplify.
-2a+6a
For any term t, t^{1}=t.
\frac{\mathrm{d}}{\mathrm{d}a}(\frac{2}{1}a^{3-1})
To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.
\frac{\mathrm{d}}{\mathrm{d}a}(2a^{2})
Do the arithmetic.
2\times 2a^{2-1}
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}.
4a^{1}
Do the arithmetic.
4a
For any term t, t^{1}=t.
2a^{2}
Cancel out a in both numerator and denominator.
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}