Solve (frac{1/2)^{-4}*6^5*2^-2*b}{6^6*2^3}

$$\frac { ( \frac { 1 } { 2 } ) ^ { - 4 } \cdot 6 ^ { 5 } \cdot 2 ^ { - 2 } \cdot b } { 6 ^ { 6 } \cdot 2 ^ { 3 } }$$

Evaluate

$$\frac{b}{12}$$

Solution Steps

Cancel out 6^{5} in both numerator and denominator.

$$\frac{\left(\frac{1}{2}\right)^{-4}\times 2^{-2}b}{6\times 2^{3}}$$

To divide powers of the same base, subtract the numerator's exponent from the denominator's exponent.

$$\frac{\left(\frac{1}{2}\right)^{-4}b}{6\times 2^{5}}$$

Calculate \frac{1}{2} to the power of -4 and get 16.

$$\frac{16b}{6\times 2^{5}}$$

Calculate 2 to the power of 5 and get 32.

$$\frac{16b}{6\times 32}$$

Multiply 6 and 32 to get 192.

$$\frac{16b}{192}$$

Divide 16b by 192 to get \frac{1}{12}b.

$$\frac{1}{12}b$$

Differentiate w.r.t. b

$$\frac{1}{12} = 0.08333333333333333$$

Steps Using Definition of a Derivative

Cancel out 6^{5} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{\left(\frac{1}{2}\right)^{-4}\times 2^{-2}b}{6\times 2^{3}})$$

To divide powers of the same base, subtract the numerator's exponent from the denominator's exponent.

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{\left(\frac{1}{2}\right)^{-4}b}{6\times 2^{5}})$$

Calculate \frac{1}{2} to the power of -4 and get 16.

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{16b}{6\times 2^{5}})$$

Calculate 2 to the power of 5 and get 32.

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{16b}{6\times 32})$$

Multiply 6 and 32 to get 192.

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{16b}{192})$$

Divide 16b by 192 to get \frac{1}{12}b.

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{1}{12}b)$$

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

$$\frac{1}{12}b^{1-1}$$

Subtract 1 from 1.

$$\frac{1}{12}b^{0}$$

For any term t except 0, t^{0}=1.

$$\frac{1}{12}\times 1$$

For any term t, t\times 1=t and 1t=t.

$$\frac{1}{12}$$