Solve 1/a+5b-a/a^2-25b^2+1/2a-10b*46

$$\frac { 1 } { a + 5 b } - \frac { a } { a ^ { 2 } - 25 b ^ { 2 } } + \frac { 1 } { 2 a - 10 b } \cdot 46$$

Evaluate

$$\frac{23a+110b}{a^{2}-25b^{2}}$$

Short Solution Steps

Express \frac{1}{2a-10b}\times 46 as a single fraction.

$$\frac{1}{a+5b}-\frac{a}{a^{2}-25b^{2}}+\frac{46}{2a-10b}$$

Factor a^{2}-25b^{2}.

$$\frac{1}{a+5b}-\frac{a}{\left(a-5b\right)\left(a+5b\right)}+\frac{46}{2a-10b}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a+5b and \left(a-5b\right)\left(a+5b\right) is \left(a-5b\right)\left(a+5b\right). Multiply \frac{1}{a+5b} times \frac{a-5b}{a-5b}.

$$\frac{a-5b}{\left(a-5b\right)\left(a+5b\right)}-\frac{a}{\left(a-5b\right)\left(a+5b\right)}+\frac{46}{2a-10b}$$

Since \frac{a-5b}{\left(a-5b\right)\left(a+5b\right)} and \frac{a}{\left(a-5b\right)\left(a+5b\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{a-5b-a}{\left(a-5b\right)\left(a+5b\right)}+\frac{46}{2a-10b}$$

Combine like terms in a-5b-a.

$$\frac{-5b}{\left(a-5b\right)\left(a+5b\right)}+\frac{46}{2a-10b}$$

Factor 2a-10b.

$$\frac{-5b}{\left(a-5b\right)\left(a+5b\right)}+\frac{46}{2\left(a-5b\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(a-5b\right)\left(a+5b\right) and 2\left(a-5b\right) is 2\left(a-5b\right)\left(a+5b\right). Multiply \frac{-5b}{\left(a-5b\right)\left(a+5b\right)} times \frac{2}{2}. Multiply \frac{46}{2\left(a-5b\right)} times \frac{a+5b}{a+5b}.

$$\frac{2\left(-5\right)b}{2\left(a-5b\right)\left(a+5b\right)}+\frac{46\left(a+5b\right)}{2\left(a-5b\right)\left(a+5b\right)}$$

Since \frac{2\left(-5\right)b}{2\left(a-5b\right)\left(a+5b\right)} and \frac{46\left(a+5b\right)}{2\left(a-5b\right)\left(a+5b\right)} have the same denominator, add them by adding their numerators.

$$\frac{2\left(-5\right)b+46\left(a+5b\right)}{2\left(a-5b\right)\left(a+5b\right)}$$

Do the multiplications in 2\left(-5\right)b+46\left(a+5b\right).

$$\frac{-10b+46a+230b}{2\left(a-5b\right)\left(a+5b\right)}$$

Combine like terms in -10b+46a+230b.

$$\frac{220b+46a}{2\left(a-5b\right)\left(a+5b\right)}$$

Factor the expressions that are not already factored in \frac{220b+46a}{2\left(a-5b\right)\left(a+5b\right)}.

$$\frac{2\left(23a+110b\right)}{2\left(a-5b\right)\left(a+5b\right)}$$

Cancel out 2 in both numerator and denominator.

$$\frac{23a+110b}{\left(a-5b\right)\left(a+5b\right)}$$

Expand \left(a-5b\right)\left(a+5b\right).

$$\frac{23a+110b}{a^{2}-25b^{2}}$$

Differentiate w.r.t. a

$$\frac{-23a^{2}-220ab-575b^{2}}{\left(a^{2}-25b^{2}\right)^{2}}$$