Solve 16x^2/frac{x^4{4}-8x^2+64}

$$\frac { 16 x ^ { 2 } } { \frac { x ^ { 4 } } { 4 } - 8 x ^ { 2 } + 64 }$$

Evaluate

$$64\times \left(\frac{x}{x^{2}-16}\right)^{2}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply -8x^{2}+64 times \frac{4}{4}.

$$\frac{16x^{2}}{\frac{x^{4}}{4}+\frac{4\left(-8x^{2}+64\right)}{4}}$$

Since \frac{x^{4}}{4} and \frac{4\left(-8x^{2}+64\right)}{4} have the same denominator, add them by adding their numerators.

$$\frac{16x^{2}}{\frac{x^{4}+4\left(-8x^{2}+64\right)}{4}}$$

Do the multiplications in x^{4}+4\left(-8x^{2}+64\right).

$$\frac{16x^{2}}{\frac{x^{4}-32x^{2}+256}{4}}$$

Divide 16x^{2} by \frac{x^{4}-32x^{2}+256}{4} by multiplying 16x^{2} by the reciprocal of \frac{x^{4}-32x^{2}+256}{4}.

$$\frac{16x^{2}\times 4}{x^{4}-32x^{2}+256}$$

Multiply 16 and 4 to get 64.

$$\frac{64x^{2}}{x^{4}-32x^{2}+256}$$

Differentiate w.r.t. x

$$\frac{128x\left(-x^{2}-16\right)}{\left(x^{2}-16\right)^{3}}$$

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.

$$\frac{\left(\frac{1}{4}x^{4}-8x^{2}+64\right)\frac{\mathrm{d}}{\mathrm{d}x}(16x^{2})-16x^{2}\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{4}x^{4}-8x^{2}+64)}{\left(\frac{1}{4}x^{4}-8x^{2}+64\right)^{2}}$$

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}.

$$\frac{\left(\frac{1}{4}x^{4}-8x^{2}+64\right)\times 2\times 16x^{2-1}-16x^{2}\left(4\times \frac{1}{4}x^{4-1}+2\left(-8\right)x^{2-1}\right)}{\left(\frac{1}{4}x^{4}-8x^{2}+64\right)^{2}}$$

Simplify.

$$\frac{\left(\frac{1}{4}x^{4}-8x^{2}+64\right)\times 32x^{1}-16x^{2}\left(x^{3}-16x^{1}\right)}{\left(\frac{1}{4}x^{4}-8x^{2}+64\right)^{2}}$$

Multiply \frac{1}{4}x^{4}-8x^{2}+64 times 32x^{1}.

$$\frac{\frac{1}{4}x^{4}\times 32x^{1}-8x^{2}\times 32x^{1}+64\times 32x^{1}-16x^{2}\left(x^{3}-16x^{1}\right)}{\left(\frac{1}{4}x^{4}-8x^{2}+64\right)^{2}}$$

Multiply 16x^{2} times x^{3}-16x^{1}.

$$\frac{\frac{1}{4}x^{4}\times 32x^{1}-8x^{2}\times 32x^{1}+64\times 32x^{1}-\left(16x^{2}x^{3}+16x^{2}\left(-16\right)x^{1}\right)}{\left(\frac{1}{4}x^{4}-8x^{2}+64\right)^{2}}$$

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

$$\frac{\frac{1}{4}\times 32x^{4+1}-8\times 32x^{2+1}+64\times 32x^{1}-\left(16x^{2+3}+16\left(-16\right)x^{2+1}\right)}{\left(\frac{1}{4}x^{4}-8x^{2}+64\right)^{2}}$$

Simplify.

$$\frac{8x^{5}-256x^{3}+2048x^{1}-\left(16x^{5}-256x^{3}\right)}{\left(\frac{1}{4}x^{4}-8x^{2}+64\right)^{2}}$$

Combine like terms.

$$\frac{-8x^{5}+2048x^{1}}{\left(\frac{1}{4}x^{4}-8x^{2}+64\right)^{2}}$$

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

$$\frac{-8x^{5}+2048x}{\left(\frac{1}{4}x^{4}-8x^{2}+64\right)^{2}}$$