Solve 8x/sqrt{2x^2+1}

$$\frac{ 8x }{ \sqrt{ 2 } { x }^{ 2 } +1 }$$

Evaluate

$$\frac{8x}{\sqrt{2}x^{2}+1}$$

Short Solution Steps

Rationalize the denominator of \frac{8x}{\sqrt{2}x^{2}+1} by multiplying numerator and denominator by \sqrt{2}x^{2}-1.

$$\frac{8x\left(\sqrt{2}x^{2}-1\right)}{\left(\sqrt{2}x^{2}+1\right)\left(\sqrt{2}x^{2}-1\right)}$$

Consider \left(\sqrt{2}x^{2}+1\right)\left(\sqrt{2}x^{2}-1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{8x\left(\sqrt{2}x^{2}-1\right)}{\left(\sqrt{2}x^{2}\right)^{2}-1^{2}}$$

Expand \left(\sqrt{2}x^{2}\right)^{2}.

$$\frac{8x\left(\sqrt{2}x^{2}-1\right)}{\left(\sqrt{2}\right)^{2}\left(x^{2}\right)^{2}-1^{2}}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$\frac{8x\left(\sqrt{2}x^{2}-1\right)}{\left(\sqrt{2}\right)^{2}x^{4}-1^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{8x\left(\sqrt{2}x^{2}-1\right)}{2x^{4}-1^{2}}$$

Calculate 1 to the power of 2 and get 1.

$$\frac{8x\left(\sqrt{2}x^{2}-1\right)}{2x^{4}-1}$$

Use the distributive property to multiply 8x by \sqrt{2}x^{2}-1.

$$\frac{8\sqrt{2}x^{3}-8x}{2x^{4}-1}$$

Differentiate w.r.t. x

$$\frac{8\left(-\sqrt{2}x^{2}+1\right)}{\left(\sqrt{2}x^{2}+1\right)^{2}}$$

Steps Using Derivative Rule for Quotient

Rationalize the denominator of \frac{8x}{\sqrt{2}x^{2}+1} by multiplying numerator and denominator by \sqrt{2}x^{2}-1.

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

Consider \left(\sqrt{2}x^{2}+1\right)\left(\sqrt{2}x^{2}-1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{8x\left(\sqrt{2}x^{2}-1\right)}{\left(\sqrt{2}x^{2}\right)^{2}-1^{2}})$$

Expand \left(\sqrt{2}x^{2}\right)^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{8x\left(\sqrt{2}x^{2}-1\right)}{\left(\sqrt{2}\right)^{2}\left(x^{2}\right)^{2}-1^{2}})$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

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

The square of \sqrt{2} is 2.

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

Calculate 1 to the power of 2 and get 1.

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

Use the distributive property to multiply 8x by \sqrt{2}x^{2}-1.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{8\sqrt{2}x^{3}-8x}{2x^{4}-1})$$

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.

$$\frac{\left(2x^{4}-1\right)\frac{\mathrm{d}}{\mathrm{d}x}(8\sqrt{2}x^{3}-8x^{1})-\left(8\sqrt{2}x^{3}-8x^{1}\right)\frac{\mathrm{d}}{\mathrm{d}x}(2x^{4}-1)}{\left(2x^{4}-1\right)^{2}}$$

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}.

$$\frac{\left(2x^{4}-1\right)\left(3\times 8\sqrt{2}x^{3-1}-8x^{1-1}\right)-\left(8\sqrt{2}x^{3}-8x^{1}\right)\times 4\times 2x^{4-1}}{\left(2x^{4}-1\right)^{2}}$$

Simplify.

$$\frac{\left(2x^{4}-1\right)\left(24\sqrt{2}x^{2}-8x^{0}\right)-\left(8\sqrt{2}x^{3}-8x^{1}\right)\times 8x^{3}}{\left(2x^{4}-1\right)^{2}}$$

Multiply 2x^{4}-1 times 24\sqrt{2}x^{2}-8x^{0}.

$$\frac{2x^{4}\times 24\sqrt{2}x^{2}+2x^{4}\left(-8\right)x^{0}-24\sqrt{2}x^{2}-\left(-8x^{0}\right)-\left(8\sqrt{2}x^{3}-8x^{1}\right)\times 8x^{3}}{\left(2x^{4}-1\right)^{2}}$$

Multiply 8\sqrt{2}x^{3}-8x^{1} times 8x^{3}.

$$\frac{2x^{4}\times 24\sqrt{2}x^{2}+2x^{4}\left(-8\right)x^{0}-24\sqrt{2}x^{2}-\left(-8x^{0}\right)-\left(8\sqrt{2}x^{3}\times 8x^{3}-8x^{1}\times 8x^{3}\right)}{\left(2x^{4}-1\right)^{2}}$$

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

$$\frac{2\times 24\sqrt{2}x^{4+2}+2\left(-8\right)x^{4}-24\sqrt{2}x^{2}-\left(-8x^{0}\right)-\left(8\sqrt{2}\times 8x^{3+3}-8\times 8x^{1+3}\right)}{\left(2x^{4}-1\right)^{2}}$$

Simplify.

$$\frac{48\sqrt{2}x^{6}-16x^{4}+\left(-24\sqrt{2}\right)x^{2}+8x^{0}-\left(64\sqrt{2}x^{6}-64x^{4}\right)}{\left(2x^{4}-1\right)^{2}}$$

Combine like terms.

$$\frac{\left(-16\sqrt{2}\right)x^{6}+48x^{4}+\left(-24\sqrt{2}\right)x^{2}+8x^{0}}{\left(2x^{4}-1\right)^{2}}$$

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

$$\frac{\left(-16\sqrt{2}\right)x^{6}+48x^{4}+\left(-24\sqrt{2}\right)x^{2}+8\times 1}{\left(2x^{4}-1\right)^{2}}$$

For any term t, t\times 1=t and 1t=t.

$$\frac{\left(-16\sqrt{2}\right)x^{6}+48x^{4}+\left(-24\sqrt{2}\right)x^{2}+8}{\left(2x^{4}-1\right)^{2}}$$