Solve frac{36x^{-2}y^{5}z^-11}{3*z^4*x^-6*x^3}

$$\frac { 36 x ^ { - 2 } y ^ { 5 } z ^ { - 11 } } { 3 \cdot z ^ { 4 } \cdot x ^ { - 6 } \cdot x ^ { 3 } }$$

Evaluate

$$\frac{12xy^{5}}{z^{15}}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add -6 and 3 to get -3.

$$\frac{36x^{-2}y^{5}z^{-11}}{3z^{4}x^{-3}}$$

Cancel out 3 in both numerator and denominator.

$$\frac{12z^{-11}x^{-2}y^{5}}{x^{-3}z^{4}}$$

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

$$\frac{12z^{-11}x^{1}y^{5}}{z^{4}}$$

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

$$\frac{12x^{1}y^{5}}{z^{15}}$$

Calculate x to the power of 1 and get x.

$$\frac{12xy^{5}}{z^{15}}$$

Differentiate w.r.t. x

$$\frac{12y^{5}}{z^{15}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{36y^{5}}{z^{11}\times 3x^{3}z^{4}}x^{-2-\left(-6\right)})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{12y^{5}}{x^{3}z^{15}}x^{4})$$

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}.

$$4\times \frac{12y^{5}}{x^{3}z^{15}}x^{4-1}$$

Do the arithmetic.

$$\frac{48y^{5}}{x^{3}z^{15}}x^{3}$$