Solve frac{18ab^{4}c^3}{12a^3b^2c^3}

$$\frac { 18 a b ^ { 4 } c ^ { 3 } } { 12 a ^ { 3 } b ^ { 2 } c ^ { 3 } }$$

Differentiate w.r.t. a

$$-\frac{3b^{2}}{a^{3}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{18c^{3}b^{4}}{12b^{2}c^{3}}a^{1-3})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{3b^{2}}{2}a^{-2})$$

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}.

$$-2\times \frac{3b^{2}}{2}a^{-2-1}$$

Do the arithmetic.

$$\left(-3b^{2}\right)a^{-3}$$

Evaluate

$$\frac{3\times \left(\frac{b}{a}\right)^{2}}{2}$$

Solution Steps

Cancel out 6ab^{2}c^{3} in both numerator and denominator.

$$\frac{3b^{2}}{2a^{2}}$$