Solve 36a^2/9a^3-36a=

$$\frac { 36 a ^ { 2 } } { 9 a ^ { 3 } - 36 a } =$$

Evaluate

$$\frac{4a}{a^{2}-4}$$

Short Solution Steps

Factor the expressions that are not already factored.

$$\frac{36a^{2}}{9a\left(a-2\right)\left(a+2\right)}$$

Cancel out 9a in both numerator and denominator.

$$\frac{4a}{\left(a-2\right)\left(a+2\right)}$$

Expand the expression.

$$\frac{4a}{a^{2}-4}$$

Differentiate w.r.t. a

$$-\frac{4\left(a^{2}+4\right)}{\left(a^{2}-4\right)^{2}}$$

Steps Using Derivative Rule for Quotient

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.

$$\frac{\left(9a^{3}-36a^{1}\right)\frac{\mathrm{d}}{\mathrm{d}a}(36a^{2})-36a^{2}\frac{\mathrm{d}}{\mathrm{d}a}(9a^{3}-36a^{1})}{\left(9a^{3}-36a^{1}\right)^{2}}$$

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}.

$$\frac{\left(9a^{3}-36a^{1}\right)\times 2\times 36a^{2-1}-36a^{2}\left(3\times 9a^{3-1}-36a^{1-1}\right)}{\left(9a^{3}-36a^{1}\right)^{2}}$$

Simplify.

$$\frac{\left(9a^{3}-36a^{1}\right)\times 72a^{1}-36a^{2}\left(27a^{2}-36a^{0}\right)}{\left(9a^{3}-36a^{1}\right)^{2}}$$

Multiply 9a^{3}-36a^{1} times 72a^{1}.

$$\frac{9a^{3}\times 72a^{1}-36a^{1}\times 72a^{1}-36a^{2}\left(27a^{2}-36a^{0}\right)}{\left(9a^{3}-36a^{1}\right)^{2}}$$

Multiply 36a^{2} times 27a^{2}-36a^{0}.

$$\frac{9a^{3}\times 72a^{1}-36a^{1}\times 72a^{1}-\left(36a^{2}\times 27a^{2}+36a^{2}\left(-36\right)a^{0}\right)}{\left(9a^{3}-36a^{1}\right)^{2}}$$

To multiply powers of the same base, add their exponents.

$$\frac{9\times 72a^{3+1}-36\times 72a^{1+1}-\left(36\times 27a^{2+2}+36\left(-36\right)a^{2}\right)}{\left(9a^{3}-36a^{1}\right)^{2}}$$

Simplify.

$$\frac{648a^{4}-2592a^{2}-\left(972a^{4}-1296a^{2}\right)}{\left(9a^{3}-36a^{1}\right)^{2}}$$

Combine like terms.

$$\frac{-324a^{4}-1296a^{2}}{\left(9a^{3}-36a^{1}\right)^{2}}$$

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

$$\frac{-324a^{4}-1296a^{2}}{\left(9a^{3}-36a\right)^{2}}$$