Solve x*(-2x^{2})*4x^3*(-8x^4)*16x^5*(-32x^6)

$$x \cdot ( - 2 x ^ { 2 } ) \cdot 4 x ^ { 3 } \cdot ( - 8 x ^ { 4 } ) \cdot 16 x ^ { 5 } \cdot ( - 32 x ^ { 6 } )$$

Evaluate

$$-32768x^{21}$$

Solution Steps

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

$$x^{3}\left(-2\right)\times 4x^{3}\left(-8\right)x^{4}\times 16x^{5}\left(-32\right)x^{6}$$

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

$$x^{6}\left(-2\right)\times 4\left(-8\right)x^{4}\times 16x^{5}\left(-32\right)x^{6}$$

To multiply powers of the same base, add their exponents. Add 6 and 4 to get 10.

$$x^{10}\left(-2\right)\times 4\left(-8\right)\times 16x^{5}\left(-32\right)x^{6}$$

To multiply powers of the same base, add their exponents. Add 10 and 5 to get 15.

$$x^{15}\left(-2\right)\times 4\left(-8\right)\times 16\left(-32\right)x^{6}$$

To multiply powers of the same base, add their exponents. Add 15 and 6 to get 21.

$$x^{21}\left(-2\right)\times 4\left(-8\right)\times 16\left(-32\right)$$

Multiply -2 and 4 to get -8.

$$x^{21}\left(-8\right)\left(-8\right)\times 16\left(-32\right)$$

Multiply -8 and -8 to get 64.

$$x^{21}\times 64\times 16\left(-32\right)$$

Multiply 64 and 16 to get 1024.

$$x^{21}\times 1024\left(-32\right)$$

Multiply 1024 and -32 to get -32768.

$$x^{21}\left(-32768\right)$$

Differentiate w.r.t. x

$$-688128x^{20}$$

Steps Using Definition of a Derivative

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{3}\left(-2\right)\times 4x^{3}\left(-8\right)x^{4}\times 16x^{5}\left(-32\right)x^{6})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{6}\left(-2\right)\times 4\left(-8\right)x^{4}\times 16x^{5}\left(-32\right)x^{6})$$

To multiply powers of the same base, add their exponents. Add 6 and 4 to get 10.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{10}\left(-2\right)\times 4\left(-8\right)\times 16x^{5}\left(-32\right)x^{6})$$

To multiply powers of the same base, add their exponents. Add 10 and 5 to get 15.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{15}\left(-2\right)\times 4\left(-8\right)\times 16\left(-32\right)x^{6})$$

To multiply powers of the same base, add their exponents. Add 15 and 6 to get 21.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{21}\left(-2\right)\times 4\left(-8\right)\times 16\left(-32\right))$$

Multiply -2 and 4 to get -8.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{21}\left(-8\right)\left(-8\right)\times 16\left(-32\right))$$

Multiply -8 and -8 to get 64.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{21}\times 64\times 16\left(-32\right))$$

Multiply 64 and 16 to get 1024.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{21}\times 1024\left(-32\right))$$

Multiply 1024 and -32 to get -32768.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{21}\left(-32768\right))$$

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

$$21\left(-32768\right)x^{21-1}$$

Multiply 21 times -32768.

$$-688128x^{21-1}$$

Subtract 1 from 21.

$$-688128x^{20}$$