Solve d/dx(8x^6-9x^10)(8x^6+9x^10)

$$\frac { d } { d x } ( 8 x ^ { 6 } - 9 x ^ { 10 } ) ( 8 x ^ { 6 } + 9 x ^ { 10 } )$$

Evaluate

$$768x^{11}-1620x^{19}$$

Steps Using Derivative Rule for Sum

Consider \left(8x^{6}-9x^{10}\right)\left(8x^{6}+9x^{10}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(8x^{6}\right)^{2}-\left(9x^{10}\right)^{2})$$

Expand \left(8x^{6}\right)^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(8^{2}\left(x^{6}\right)^{2}-\left(9x^{10}\right)^{2})$$

To raise a power to another power, multiply the exponents. Multiply 6 and 2 to get 12.

$$\frac{\mathrm{d}}{\mathrm{d}x}(8^{2}x^{12}-\left(9x^{10}\right)^{2})$$

Calculate 8 to the power of 2 and get 64.

$$\frac{\mathrm{d}}{\mathrm{d}x}(64x^{12}-\left(9x^{10}\right)^{2})$$

Expand \left(9x^{10}\right)^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(64x^{12}-9^{2}\left(x^{10}\right)^{2})$$

To raise a power to another power, multiply the exponents. Multiply 10 and 2 to get 20.

$$\frac{\mathrm{d}}{\mathrm{d}x}(64x^{12}-9^{2}x^{20})$$

Calculate 9 to the power of 2 and get 81.

$$\frac{\mathrm{d}}{\mathrm{d}x}(64x^{12}-81x^{20})$$

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}.

$$12\times 64x^{12-1}+20\left(-81\right)x^{20-1}$$

Multiply 12 times 64.

$$768x^{12-1}+20\left(-81\right)x^{20-1}$$

Subtract 1 from 12.

$$768x^{11}+20\left(-81\right)x^{20-1}$$

Multiply 20 times -81.

$$768x^{11}-1620x^{20-1}$$

Subtract 1 from 20.

$$768x^{11}-1620x^{19}$$

Differentiate w.r.t. x

$$8448x^{10}-30780x^{18}$$