Solve frac{3x^{-2}x^{3}y^4}{6^-1x^-3y^0}

$$\frac { 3 x ^ { - 2 } x ^ { 3 } y ^ { 4 } } { 6 ^ { - 1 } x ^ { - 3 } y ^ { 0 } }$$

Evaluate

$$18\left(xy\right)^{4}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add -2 and 3 to get 1.

$$\frac{3x^{1}y^{4}}{6^{-1}x^{-3}y^{0}}$$

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

$$\frac{3x^{4}y^{4}}{\frac{1}{6}}$$

Divide 3x^{4}y^{4} by \frac{1}{6} by multiplying 3x^{4}y^{4} by the reciprocal of \frac{1}{6}.

$$3x^{4}y^{4}\times 6$$

Multiply 3 and 6 to get 18.

$$18x^{4}y^{4}$$

Differentiate w.r.t. x

$$72x^{3}y^{4}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{6\times 3x^{3}y^{4}}{y^{0}}x^{-2-\left(-3\right)})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(18x^{3}y^{4}x^{1})$$

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}.

$$18x^{3}y^{4}x^{1-1}$$

Do the arithmetic.

$$18x^{3}y^{4}x^{0}$$

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

$$18x^{3}y^{4}\times 1$$

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

$$18x^{3}y^{4}$$