Solve (16/81x^16)^1/2

$$( \frac { 16 } { 81 } x ^ { 16 } ) ^ { 1 / 2 }$$

Evaluate

$$\frac{4x^{8}}{9}$$

Solution Steps

Expand \left(\frac{16}{81}x^{16}\right)^{\frac{1}{2}}.

$$\left(\frac{16}{81}\right)^{\frac{1}{2}}\left(x^{16}\right)^{\frac{1}{2}}$$

To raise a power to another power, multiply the exponents. Multiply 16 and \frac{1}{2} to get 8.

$$\left(\frac{16}{81}\right)^{\frac{1}{2}}x^{8}$$

Calculate \frac{16}{81} to the power of \frac{1}{2} and get \frac{4}{9}.

$$\frac{4}{9}x^{8}$$

Differentiate w.r.t. x

$$\frac{32x^{7}}{9}$$

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).

$$\frac{1}{2}\times \left(\frac{16}{81}x^{16}\right)^{\frac{1}{2}-1}\frac{\mathrm{d}}{\mathrm{d}x}(\frac{16}{81}x^{16})$$

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}.

$$\frac{1}{2}\times \left(\frac{16}{81}x^{16}\right)^{-\frac{1}{2}}\times 16\times \frac{16}{81}x^{16-1}$$

Simplify.

$$\frac{128}{81}x^{15}\times \left(\frac{16}{81}x^{16}\right)^{-\frac{1}{2}}$$