Solve p(x)x^4/x^4+5x^2+4

$$p ( x ) \frac { x ^ { 4 } } { x ^ { 4 } + 5 x ^ { 2 } + 4 }$$

Evaluate

$$\frac{px^{5}}{x^{4}+5x^{2}+4}$$

Short Solution Steps

Express p\times \frac{x^{4}}{x^{4}+5x^{2}+4} as a single fraction.

$$\frac{px^{4}}{x^{4}+5x^{2}+4}x$$

Express \frac{px^{4}}{x^{4}+5x^{2}+4}x as a single fraction.

$$\frac{px^{4}x}{x^{4}+5x^{2}+4}$$

To multiply powers of the same base, add their exponents. Add 4 and 1 to get 5.

$$\frac{px^{5}}{x^{4}+5x^{2}+4}$$

Differentiate w.r.t. x

$$\frac{px^{4}\left(x^{4}+15x^{2}+20\right)}{x^{8}+10x^{6}+33x^{4}+40x^{2}+16}$$

Steps Using Derivative Rule for Quotient

Express p\times \frac{x^{4}}{x^{4}+5x^{2}+4} as a single fraction.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{px^{4}}{x^{4}+5x^{2}+4}x)$$

Express \frac{px^{4}}{x^{4}+5x^{2}+4}x as a single fraction.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{px^{4}x}{x^{4}+5x^{2}+4})$$

To multiply powers of the same base, add their exponents. Add 4 and 1 to get 5.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{px^{5}}{x^{4}+5x^{2}+4})$$

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.

$$\frac{\left(x^{4}+5x^{2}+4\right)\frac{\mathrm{d}}{\mathrm{d}x}(px^{5})-px^{5}\frac{\mathrm{d}}{\mathrm{d}x}(x^{4}+5x^{2}+4)}{\left(x^{4}+5x^{2}+4\right)^{2}}$$

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}.

$$\frac{\left(x^{4}+5x^{2}+4\right)\times 5px^{5-1}-px^{5}\left(4x^{4-1}+2\times 5x^{2-1}\right)}{\left(x^{4}+5x^{2}+4\right)^{2}}$$

Simplify.

$$\frac{\left(x^{4}+5x^{2}+4\right)\times 5px^{4}-px^{5}\left(4x^{3}+10x^{1}\right)}{\left(x^{4}+5x^{2}+4\right)^{2}}$$

Multiply x^{4}+5x^{2}+4 times 5px^{4}.

$$\frac{x^{4}\times 5px^{4}+5x^{2}\times 5px^{4}+4\times 5px^{4}-px^{5}\left(4x^{3}+10x^{1}\right)}{\left(x^{4}+5x^{2}+4\right)^{2}}$$

Multiply px^{5} times 4x^{3}+10x^{1}.

$$\frac{x^{4}\times 5px^{4}+5x^{2}\times 5px^{4}+4\times 5px^{4}-\left(px^{5}\times 4x^{3}+px^{5}\times 10x^{1}\right)}{\left(x^{4}+5x^{2}+4\right)^{2}}$$

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

$$\frac{5px^{4+4}+5\times 5px^{2+4}+4\times 5px^{4}-\left(p\times 4x^{5+3}+p\times 10x^{5+1}\right)}{\left(x^{4}+5x^{2}+4\right)^{2}}$$

Simplify.

$$\frac{5px^{8}+25px^{6}+20px^{4}-\left(4px^{8}+10px^{6}\right)}{\left(x^{4}+5x^{2}+4\right)^{2}}$$

Combine like terms.

$$\frac{px^{8}+15px^{6}+20px^{4}}{\left(x^{4}+5x^{2}+4\right)^{2}}$$