Solve 3y^-3/2y^2*3y^4

$$\frac { 3 y ^ { - 3 } } { 2 y ^ { 2 } \cdot 3 y ^ { 4 } }$$

Evaluate

$$\frac{1}{2y^{9}}$$

Solution Steps

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

$$\frac{3y^{-3}}{2y^{6}\times 3}$$

Cancel out 3 in both numerator and denominator.

$$\frac{y^{-3}}{2y^{6}}$$

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

$$\frac{1}{2y^{9}}$$

Differentiate w.r.t. y

$$-\frac{9}{2y^{10}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{3}{6y^{4}}y^{-3-2})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{1}{2y^{4}}y^{-5})$$

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}.

$$-5\times \frac{1}{2y^{4}}y^{-5-1}$$

Do the arithmetic.

$$\left(-\frac{5}{2y^{4}}\right)y^{-6}$$