Solve frac{9a^{7}b^{5}c^5}{18a^5b^9c^3}

$$\frac { 9 a ^ { 7 } b ^ { 5 } c ^ { 5 } } { 18 a ^ { 5 } b ^ { 9 } c ^ { 3 } }$$

Differentiate w.r.t. a

$$\frac{ac^{2}}{b^{4}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{9b^{5}c^{5}}{18c^{3}b^{9}}a^{7-5})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{c^{2}}{2b^{4}}a^{2})$$

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}.

$$2\times \frac{c^{2}}{2b^{4}}a^{2-1}$$

Do the arithmetic.

$$\frac{c^{2}}{b^{4}}a^{1}$$

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

$$\frac{c^{2}}{b^{4}}a$$

Evaluate

$$\frac{\left(ac\right)^{2}}{2b^{4}}$$

Solution Steps

Cancel out 9c^{3}a^{5}b^{5} in both numerator and denominator.

$$\frac{a^{2}c^{2}}{2b^{4}}$$