Solve (2x^4:12^4*2^-2:2*2^2*x)

$$( 2 x ^ { 4 } : 12 ^ { 4 } \cdot 2 ^ { - 2 } : 2 \cdot 2 ^ { 2 } \cdot x )$$

Evaluate

$$\frac{x^{5}}{20736}$$

Solution Steps

Calculate 12 to the power of 4 and get 20736.

$$\frac{\frac{2x^{4}}{20736}\times 2^{-2}}{2}\times 2^{2}x$$

Divide 2x^{4} by 20736 to get \frac{1}{10368}x^{4}.

$$\frac{\frac{1}{10368}x^{4}\times 2^{-2}}{2}\times 2^{2}x$$

Calculate 2 to the power of -2 and get \frac{1}{4}.

$$\frac{\frac{1}{10368}x^{4}\times \frac{1}{4}}{2}\times 2^{2}x$$

Multiply \frac{1}{10368} and \frac{1}{4} to get \frac{1}{41472}.

$$\frac{\frac{1}{41472}x^{4}}{2}\times 2^{2}x$$

Divide \frac{1}{41472}x^{4} by 2 to get \frac{1}{82944}x^{4}.

$$\frac{1}{82944}x^{4}\times 2^{2}x$$

Calculate 2 to the power of 2 and get 4.

$$\frac{1}{82944}x^{4}\times 4x$$

Multiply \frac{1}{82944} and 4 to get \frac{1}{20736}.

$$\frac{1}{20736}x^{4}x$$

To multiply powers of the same base, add their exponents. Add 4 and 1 to get 5.

$$\frac{1}{20736}x^{5}$$

Differentiate w.r.t. x

$$\frac{5x^{4}}{20736}$$

Steps Using Definition of a Derivative

Calculate 12 to the power of 4 and get 20736.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\frac{2x^{4}}{20736}\times 2^{-2}}{2}\times 2^{2}x)$$

Divide 2x^{4} by 20736 to get \frac{1}{10368}x^{4}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\frac{1}{10368}x^{4}\times 2^{-2}}{2}\times 2^{2}x)$$

Calculate 2 to the power of -2 and get \frac{1}{4}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\frac{1}{10368}x^{4}\times \frac{1}{4}}{2}\times 2^{2}x)$$

Multiply \frac{1}{10368} and \frac{1}{4} to get \frac{1}{41472}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\frac{1}{41472}x^{4}}{2}\times 2^{2}x)$$

Divide \frac{1}{41472}x^{4} by 2 to get \frac{1}{82944}x^{4}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{82944}x^{4}\times 2^{2}x)$$

Calculate 2 to the power of 2 and get 4.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{82944}x^{4}\times 4x)$$

Multiply \frac{1}{82944} and 4 to get \frac{1}{20736}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{20736}x^{4}x)$$

To multiply powers of the same base, add their exponents. Add 4 and 1 to get 5.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{20736}x^{5})$$

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

$$5\times \frac{1}{20736}x^{5-1}$$

Multiply 5 times \frac{1}{20736}.

$$\frac{5}{20736}x^{5-1}$$

Subtract 1 from 5.

$$\frac{5}{20736}x^{4}$$