Solve [(2x^6-5x^3)5/2x^4+1]^prime

$$[ \frac { ( 2 x ^ { 6 } - 5 x ^ { 3 } ) 5 } { 2 x ^ { 4 } + 1 } ] ^ { \prime }$$

Evaluate

$$5\times \left(\frac{x}{2x^{4}+1}\right)^{2}\left(8x^{7}+10x^{4}+12x^{3}-15\right)$$

Steps Using Derivative Rule for Quotient

Use the distributive property to multiply 2x^{6}-5x^{3} by 5.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{10x^{6}-25x^{3}}{2x^{4}+1})$$

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.

$$\frac{\left(2x^{4}+1\right)\frac{\mathrm{d}}{\mathrm{d}x}(10x^{6}-25x^{3})-\left(10x^{6}-25x^{3}\right)\frac{\mathrm{d}}{\mathrm{d}x}(2x^{4}+1)}{\left(2x^{4}+1\right)^{2}}$$

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}.

$$\frac{\left(2x^{4}+1\right)\left(6\times 10x^{6-1}+3\left(-25\right)x^{3-1}\right)-\left(10x^{6}-25x^{3}\right)\times 4\times 2x^{4-1}}{\left(2x^{4}+1\right)^{2}}$$

Simplify.

$$\frac{\left(2x^{4}+1\right)\left(60x^{5}-75x^{2}\right)-\left(10x^{6}-25x^{3}\right)\times 8x^{3}}{\left(2x^{4}+1\right)^{2}}$$

Multiply 2x^{4}+1 times 60x^{5}-75x^{2}.

$$\frac{2x^{4}\times 60x^{5}+2x^{4}\left(-75\right)x^{2}+60x^{5}-75x^{2}-\left(10x^{6}-25x^{3}\right)\times 8x^{3}}{\left(2x^{4}+1\right)^{2}}$$

Multiply 10x^{6}-25x^{3} times 8x^{3}.

$$\frac{2x^{4}\times 60x^{5}+2x^{4}\left(-75\right)x^{2}+60x^{5}-75x^{2}-\left(10x^{6}\times 8x^{3}-25x^{3}\times 8x^{3}\right)}{\left(2x^{4}+1\right)^{2}}$$

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

$$\frac{2\times 60x^{4+5}+2\left(-75\right)x^{4+2}+60x^{5}-75x^{2}-\left(10\times 8x^{6+3}-25\times 8x^{3+3}\right)}{\left(2x^{4}+1\right)^{2}}$$

Simplify.

$$\frac{120x^{9}-150x^{6}+60x^{5}-75x^{2}-\left(80x^{9}-200x^{6}\right)}{\left(2x^{4}+1\right)^{2}}$$

Combine like terms.

$$\frac{40x^{9}+50x^{6}+60x^{5}-75x^{2}}{\left(2x^{4}+1\right)^{2}}$$

Differentiate w.r.t. x

$$\frac{10x\left(8x^{11}-20x^{8}+120x^{4}+30x^{3}-15\right)}{\left(2x^{4}+1\right)^{3}}$$