Solve frac{(x^{3}y^{6})^frac{1{4}}}{(x^-7y^-3)^-frac{1{2}}}

$$\frac { ( x ^ { 3 } y ^ { 6 } ) ^ { \frac { 1 } { 4 } } } { ( x ^ { - 7 } y ^ { - 3 } ) ^ { - \frac { 1 } { 2 } } }$$

Differentiate w.r.t. x

$$-\frac{11}{4x^{\frac{15}{4}}}$$

Steps Using Chain Rule

Expand \left(x^{3}y^{6}\right)^{\frac{1}{4}}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\left(x^{3}\right)^{\frac{1}{4}}\left(y^{6}\right)^{\frac{1}{4}}}{\left(x^{-7}y^{-3}\right)^{-\frac{1}{2}}})$$

To raise a power to another power, multiply the exponents. Multiply 3 and \frac{1}{4} to get \frac{3}{4}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{\frac{3}{4}}\left(y^{6}\right)^{\frac{1}{4}}}{\left(x^{-7}y^{-3}\right)^{-\frac{1}{2}}})$$

To raise a power to another power, multiply the exponents. Multiply 6 and \frac{1}{4} to get \frac{3}{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{\frac{3}{4}}y^{\frac{3}{2}}}{\left(x^{-7}y^{-3}\right)^{-\frac{1}{2}}})$$

Expand \left(x^{-7}y^{-3}\right)^{-\frac{1}{2}}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{\frac{3}{4}}y^{\frac{3}{2}}}{\left(x^{-7}\right)^{-\frac{1}{2}}\left(y^{-3}\right)^{-\frac{1}{2}}})$$

To raise a power to another power, multiply the exponents. Multiply -7 and -\frac{1}{2} to get \frac{7}{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{\frac{3}{4}}y^{\frac{3}{2}}}{x^{\frac{7}{2}}\left(y^{-3}\right)^{-\frac{1}{2}}})$$

To raise a power to another power, multiply the exponents. Multiply -3 and -\frac{1}{2} to get \frac{3}{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{\frac{3}{4}}y^{\frac{3}{2}}}{x^{\frac{7}{2}}y^{\frac{3}{2}}})$$

Cancel out x^{\frac{3}{4}}y^{\frac{3}{2}} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{x^{\frac{11}{4}}})$$

If F is the composition of two differentiable functions f\left(u\right) and u=g\left(x\right), that is, if F\left(x\right)=f\left(g\left(x\right)\right), then the derivative of F is the derivative of f with respect to u times the derivative of g with respect to x, that is, \frac{\mathrm{d}}{\mathrm{d}x}(F)\left(x\right)=\frac{\mathrm{d}}{\mathrm{d}x}(f)\left(g\left(x\right)\right)\frac{\mathrm{d}}{\mathrm{d}x}(g)\left(x\right).

$$-\left(x^{\frac{11}{4}}\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}x}(x^{\frac{11}{4}})$$

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}.

$$-\left(x^{\frac{11}{4}}\right)^{-2}\times \frac{11}{4}x^{\frac{11}{4}-1}$$

Simplify.

$$-\frac{11}{4}x^{\frac{7}{4}}\left(x^{\frac{11}{4}}\right)^{-2}$$

Evaluate

$$\frac{1}{x^{\frac{11}{4}}}$$