Solve frac{9a^{4}b^2c^-6*4ab^4c^0}{3ab}

$$\frac { 9 a ^ { 4 } b ^ { 2 } c ^ { - 6 } \times 4 a b ^ { 4 } c ^ { 0 } } { 3 a b }$$

Evaluate

$$\frac{12a^{4}b^{5}}{c^{6}}$$

Solution Steps

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

$$\frac{9a^{5}b^{2}c^{-6}\times 4b^{4}c^{0}}{3ab}$$

To multiply powers of the same base, add their exponents. Add 2 and 4 to get 6.

$$\frac{9a^{5}b^{6}c^{-6}\times 4c^{0}}{3ab}$$

To multiply powers of the same base, add their exponents. Add -6 and 0 to get -6.

$$\frac{9a^{5}b^{6}c^{-6}\times 4}{3ab}$$

Cancel out 3ab in both numerator and denominator.

$$3\times 4c^{-6}a^{4}b^{5}$$

Multiply 3 and 4 to get 12.

$$12c^{-6}a^{4}b^{5}$$

Differentiate w.r.t. a

$$\frac{48a^{3}b^{5}}{c^{6}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{36ab^{6}}{c^{6}\times 3b}a^{4-1})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{12ab^{5}}{c^{6}}a^{3})$$

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}.

$$3\times \frac{12ab^{5}}{c^{6}}a^{3-1}$$

Do the arithmetic.

$$\frac{36ab^{5}}{c^{6}}a^{2}$$