Solve (9x^2arctan(3x-1)/6(9x^2-6x+2))^prime

$$( \frac { 9 x ^ { 2 } \arctan ( 3 x - 1 ) } { 6 ( 9 x ^ { 2 } - 6 x + 2 ) } ) ^ { \prime }$$

Evaluate

$$\frac{3x\left(-6x\arctan(3x-1)+4\arctan(3x-1)+3x\right)}{2\left(9x^{2}-6x+2\right)^{2}}$$

Steps Using Derivative Rule for Quotient

Cancel out 3 in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{3x^{2}\arctan(3x-1)}{2\left(9x^{2}-6x+2\right)})$$

Use the distributive property to multiply 2 by 9x^{2}-6x+2.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{3x^{2}\arctan(3x-1)}{18x^{2}-12x+4})$$

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.

$$\frac{\left(18x^{2}-12x^{1}+4\right)\frac{\mathrm{d}}{\mathrm{d}x}(3\arctan(3x-1)x^{2})-3\arctan(3x-1)x^{2}\frac{\mathrm{d}}{\mathrm{d}x}(18x^{2}-12x^{1}+4)}{\left(18x^{2}-12x^{1}+4\right)^{2}}$$

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}.

$$\frac{\left(18x^{2}-12x^{1}+4\right)\times 2\times 3\arctan(3x-1)x^{2-1}-3\arctan(3x-1)x^{2}\left(2\times 18x^{2-1}-12x^{1-1}\right)}{\left(18x^{2}-12x^{1}+4\right)^{2}}$$

Simplify.

$$\frac{\left(18x^{2}-12x^{1}+4\right)\times 6\arctan(3x-1)x^{1}-3\arctan(3x-1)x^{2}\left(36x^{1}-12x^{0}\right)}{\left(18x^{2}-12x^{1}+4\right)^{2}}$$

Multiply 18x^{2}-12x^{1}+4 times 6\arctan(3x-1)x^{1}.

$$\frac{18x^{2}\times 6\arctan(3x-1)x^{1}-12x^{1}\times 6\arctan(3x-1)x^{1}+4\times 6\arctan(3x-1)x^{1}-3\arctan(3x-1)x^{2}\left(36x^{1}-12x^{0}\right)}{\left(18x^{2}-12x^{1}+4\right)^{2}}$$

Multiply 3\arctan(3x-1)x^{2} times 36x^{1}-12x^{0}.

$$\frac{18x^{2}\times 6\arctan(3x-1)x^{1}-12x^{1}\times 6\arctan(3x-1)x^{1}+4\times 6\arctan(3x-1)x^{1}-\left(3\arctan(3x-1)x^{2}\times 36x^{1}+3\arctan(3x-1)x^{2}\left(-12\right)x^{0}\right)}{\left(18x^{2}-12x^{1}+4\right)^{2}}$$

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

$$\frac{18\times 6\arctan(3x-1)x^{2+1}-12\times 6\arctan(3x-1)x^{1+1}+4\times 6\arctan(3x-1)x^{1}-\left(3\arctan(3x-1)\times 36x^{2+1}+3\arctan(3x-1)\left(-12\right)x^{2}\right)}{\left(18x^{2}-12x^{1}+4\right)^{2}}$$

Simplify.

$$\frac{108\arctan(3x-1)x^{3}+\left(-72\arctan(3x-1)\right)x^{2}+24\arctan(3x-1)x^{1}-\left(108\arctan(3x-1)x^{3}+\left(-36\arctan(3x-1)\right)x^{2}\right)}{\left(18x^{2}-12x^{1}+4\right)^{2}}$$

Combine like terms.

$$\frac{\left(-36\arctan(3x-1)\right)x^{2}+24\arctan(3x-1)x^{1}}{\left(18x^{2}-12x^{1}+4\right)^{2}}$$

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

$$\frac{\left(-36\arctan(3x-1)\right)x^{2}+24\arctan(3x-1)x}{\left(18x^{2}-12x+4\right)^{2}}$$

Differentiate w.r.t. x

$$\frac{3\left(54x^{3}\arctan(3x-1)-54x^{2}\arctan(3x-1)+4\arctan(3x-1)-27x^{3}-9x^{2}+12x\right)}{\left(9x^{2}-6x+2\right)^{3}}$$