Solve 5x^3*4x:10x^2-16x^3:4x^2*2x

$$5 x ^ { 3 } \cdot 4 x : 10 x ^ { 2 } - 16 x ^ { 3 } : 4 x ^ { 2 } \cdot 2 x$$

Evaluate

$$-6x^{6}$$

Solution Steps

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

$$\frac{5x^{4}\times 4}{10}x^{2}-\frac{16x^{3}}{4}x^{2}\times 2x$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$\frac{5x^{4}\times 4}{10}x^{2}-\frac{16x^{3}}{4}x^{3}\times 2$$

Multiply 5 and 4 to get 20.

$$\frac{20x^{4}}{10}x^{2}-\frac{16x^{3}}{4}x^{3}\times 2$$

Divide 20x^{4} by 10 to get 2x^{4}.

$$2x^{4}x^{2}-\frac{16x^{3}}{4}x^{3}\times 2$$

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

$$2x^{6}-\frac{16x^{3}}{4}x^{3}\times 2$$

Divide 16x^{3} by 4 to get 4x^{3}.

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

To multiply powers of the same base, add their exponents. Add 3 and 3 to get 6.

$$2x^{6}-4x^{6}\times 2$$

Multiply 4 and 2 to get 8.

$$2x^{6}-8x^{6}$$

Combine 2x^{6} and -8x^{6} to get -6x^{6}.

$$-6x^{6}$$

Differentiate w.r.t. x

$$-36x^{5}$$

Steps Using Definition of a Derivative

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{5x^{4}\times 4}{10}x^{2}-\frac{16x^{3}}{4}x^{2}\times 2x)$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{5x^{4}\times 4}{10}x^{2}-\frac{16x^{3}}{4}x^{3}\times 2)$$

Multiply 5 and 4 to get 20.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{20x^{4}}{10}x^{2}-\frac{16x^{3}}{4}x^{3}\times 2)$$

Divide 20x^{4} by 10 to get 2x^{4}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(2x^{4}x^{2}-\frac{16x^{3}}{4}x^{3}\times 2)$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(2x^{6}-\frac{16x^{3}}{4}x^{3}\times 2)$$

Divide 16x^{3} by 4 to get 4x^{3}.

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

To multiply powers of the same base, add their exponents. Add 3 and 3 to get 6.

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

Multiply 4 and 2 to get 8.

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

Combine 2x^{6} and -8x^{6} to get -6x^{6}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(-6x^{6})$$

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

$$6\left(-6\right)x^{6-1}$$

Multiply 6 times -6.

$$-36x^{6-1}$$

Subtract 1 from 6.

$$-36x^{5}$$