Solve (frac{b^{-2}a^{-2}t^0}{b^-3a^-2t^1})

$$( \frac { b ^ { - 2 } a ^ { - 2 } t ^ { 0 } } { b ^ { - 3 } a ^ { - 2 } t ^ { 1 } } )$$

Evaluate

$$\frac{b}{t}$$

Solution Steps

Cancel out a^{-2} in both numerator and denominator.

$$\frac{b^{-2}t^{0}}{b^{-3}t^{1}}$$

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

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

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

$$\frac{b^{1}}{t^{1}}$$

Calculate b to the power of 1 and get b.

$$\frac{b}{t^{1}}$$

Calculate t to the power of 1 and get t.

$$\frac{b}{t}$$

Differentiate w.r.t. b

$$\frac{1}{t}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{t^{0}}{a^{2}\times \frac{t^{1}}{a^{2}}}b^{-2-\left(-3\right)})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{1}{t}b^{1})$$

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

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

Do the arithmetic.

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

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

$$\frac{1}{t}\times 1$$

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

$$\frac{1}{t}$$