Solve frac{(5x^frac{1{9}})(3x^frac{2{5}})}{15x^frac{5{2}+2}}=

$$\frac { ( 5 x ^ { \frac { 1 } { 9 } } ) ( 3 x ^ { \frac { 2 } { 5 } } ) } { 15 x ^ { \frac { 5 } { 2 } + 2 } } =$$

Differentiate w.r.t. x

$$-\frac{359}{90x^{\frac{449}{90}}}$$

Steps Using Chain Rule

To multiply powers of the same base, add their exponents. Add \frac{1}{9} and \frac{2}{5} to get \frac{23}{45}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{5x^{\frac{23}{45}}\times 3}{15x^{\frac{5}{2}+2}})$$

Cancel out 3\times 5 in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{\frac{23}{45}}}{x^{2+\frac{5}{2}}})$$

Add 2 and \frac{5}{2} to get \frac{9}{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{\frac{23}{45}}}{x^{\frac{9}{2}}})$$

Cancel out x^{\frac{23}{45}} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{x^{\frac{359}{90}}})$$

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).

$$-\left(x^{\frac{359}{90}}\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}x}(x^{\frac{359}{90}})$$

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}.

$$-\left(x^{\frac{359}{90}}\right)^{-2}\times \frac{359}{90}x^{\frac{359}{90}-1}$$

Simplify.

$$-\frac{359}{90}x^{\frac{269}{90}}\left(x^{\frac{359}{90}}\right)^{-2}$$

Evaluate

$$\frac{1}{x^{\frac{359}{90}}}$$