Solve 6x*2/3x^3*3x^2*2x^2/3

$$\frac { 6 x \cdot \frac { 2 } { 3 } x ^ { 3 } \cdot 3 x ^ { 2 } \cdot 2 x ^ { 2 } } { 3 }$$

Evaluate

$$8x^{8}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 1 and 3 to get 4.

$$\frac{6x^{4}\times \frac{2}{3}\times 3x^{2}\times 2x^{2}}{3}$$

To multiply powers of the same base, add their exponents. Add 4 and 2 to get 6.

$$\frac{6x^{6}\times \frac{2}{3}\times 3\times 2x^{2}}{3}$$

To multiply powers of the same base, add their exponents. Add 6 and 2 to get 8.

$$\frac{6x^{8}\times \frac{2}{3}\times 3\times 2}{3}$$

Cancel out 3 and 3.

$$6x^{8}\times \frac{2}{3}\times 2$$

Multiply 6 and \frac{2}{3} to get 4.

$$4x^{8}\times 2$$

Multiply 4 and 2 to get 8.

$$8x^{8}$$

Differentiate w.r.t. x

$$64x^{7}$$

Steps Using Definition of a Derivative

To multiply powers of the same base, add their exponents. Add 1 and 3 to get 4.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{6x^{4}\times \frac{2}{3}\times 3x^{2}\times 2x^{2}}{3})$$

To multiply powers of the same base, add their exponents. Add 4 and 2 to get 6.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{6x^{6}\times \frac{2}{3}\times 3\times 2x^{2}}{3})$$

To multiply powers of the same base, add their exponents. Add 6 and 2 to get 8.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{6x^{8}\times \frac{2}{3}\times 3\times 2}{3})$$

Cancel out 3 and 3.

$$\frac{\mathrm{d}}{\mathrm{d}x}(6x^{8}\times \frac{2}{3}\times 2)$$

Multiply 6 and \frac{2}{3} to get 4.

$$\frac{\mathrm{d}}{\mathrm{d}x}(4x^{8}\times 2)$$

Multiply 4 and 2 to get 8.

$$\frac{\mathrm{d}}{\mathrm{d}x}(8x^{8})$$

The derivative of ax^{n} is nax^{n-1}.

$$8\times 8x^{8-1}$$

Multiply 8 times 8.

$$64x^{8-1}$$

Subtract 1 from 8.

$$64x^{7}$$