Solve 3/a-3+2/3-a

$$\frac { 3 } { a - 3 } + \frac { 2 } { 3 - a }$$

Evaluate

$$\frac{1}{a-3}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a-3 and 3-a is a-3. Multiply \frac{2}{3-a} times \frac{-1}{-1}.

$$\frac{3}{a-3}+\frac{2\left(-1\right)}{a-3}$$

Since \frac{3}{a-3} and \frac{2\left(-1\right)}{a-3} have the same denominator, add them by adding their numerators.

$$\frac{3+2\left(-1\right)}{a-3}$$

Do the multiplications in 3+2\left(-1\right).

$$\frac{3-2}{a-3}$$

Do the calculations in 3-2.

$$\frac{1}{a-3}$$

Differentiate w.r.t. a

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

Steps Using Chain Rule

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a-3 and 3-a is a-3. Multiply \frac{2}{3-a} times \frac{-1}{-1}.

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

Since \frac{3}{a-3} and \frac{2\left(-1\right)}{a-3} have the same denominator, add them by adding their numerators.

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

Do the multiplications in 3+2\left(-1\right).

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{3-2}{a-3})$$

Do the calculations in 3-2.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{1}{a-3})$$

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).

$$-\left(a^{1}-3\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}a}(a^{1}-3)$$

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}.

$$-\left(a^{1}-3\right)^{-2}a^{1-1}$$

Simplify.

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

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

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

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

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