Solve frac{4x^{2}y^2}{6x^6y^2*4x^5}

$$\frac { 4 x ^ { 2 } y ^ { 2 } } { 6 x ^ { 6 } y ^ { 2 } \cdot 4 x ^ { 5 } }$$

Evaluate

$$\frac{1}{6x^{9}}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 6 and 5 to get 11.

$$\frac{4x^{2}y^{2}}{6x^{11}y^{2}\times 4}$$

Cancel out 2\times 2x^{2}y^{2} in both numerator and denominator.

$$\frac{1}{2\times 3x^{9}}$$

Multiply 2 and 3 to get 6.

$$\frac{1}{6x^{9}}$$

Differentiate w.r.t. x

$$-\frac{3}{2x^{10}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{4y^{2}}{24y^{2}x^{5}}x^{2-6})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{6x^{5}}x^{-4})$$

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}.

$$-4\times \frac{1}{6x^{5}}x^{-4-1}$$

Do the arithmetic.

$$\left(-\frac{2}{3x^{5}}\right)x^{-5}$$