Solve 6x^10y^4/3x^8x^7

$$\frac { 6 x ^ { 10 } y ^ { 4 } } { 3 x ^ { 8 } x ^ { 7 } }$$

Evaluate

$$\frac{2y^{4}}{x^{5}}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 8 and 7 to get 15.

$$\frac{6x^{10}y^{4}}{3x^{15}}$$

Cancel out 3x^{10} in both numerator and denominator.

$$\frac{2y^{4}}{x^{5}}$$

Differentiate w.r.t. x

$$-\frac{10y^{4}}{x^{6}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{6y^{4}}{3x^{7}}x^{10-8})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2y^{4}}{x^{7}}x^{2})$$

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}.

$$2\times \frac{2y^{4}}{x^{7}}x^{2-1}$$

Do the arithmetic.

$$\frac{4y^{4}}{x^{7}}x^{1}$$

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

$$\frac{4y^{4}}{x^{7}}x$$