Solve (x^frac{5{6}})^9

$$( x ^ { \frac { 5 } { 6 } } ) ^ { 9 }$$

Differentiate w.r.t. x

$$\frac{15x^{\frac{13}{2}}}{2}$$

Steps Using Chain Rule

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).

$$9\left(x^{\frac{5}{6}}\right)^{9-1}\frac{\mathrm{d}}{\mathrm{d}x}(x^{\frac{5}{6}})$$

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}.

$$9\left(x^{\frac{5}{6}}\right)^{8}\times \frac{5}{6}x^{\frac{5}{6}-1}$$

Simplify.

$$\frac{15}{2}x^{-\frac{1}{6}}\left(x^{\frac{5}{6}}\right)^{8}$$

Evaluate

$$x^{\frac{15}{2}}$$