Solve (216x^-2)^-frac{4{3}}

$$( 216 x ^ { - 2 } ) ^ { - \frac { 4 } { 3 } }$$

Evaluate

$$\frac{x^{\frac{8}{3}}}{1296}$$

Solution Steps

Expand \left(216x^{-2}\right)^{-\frac{4}{3}}.

$$216^{-\frac{4}{3}}\left(x^{-2}\right)^{-\frac{4}{3}}$$

To raise a power to another power, multiply the exponents. Multiply -2 and -\frac{4}{3} to get \frac{8}{3}.

$$216^{-\frac{4}{3}}x^{\frac{8}{3}}$$

Calculate 216 to the power of -\frac{4}{3} and get \frac{1}{1296}.

$$\frac{1}{1296}x^{\frac{8}{3}}$$

Differentiate w.r.t. x

$$\frac{x^{\frac{5}{3}}}{486}$$

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).

$$-\frac{4}{3}\times \left(216x^{-2}\right)^{-\frac{4}{3}-1}\frac{\mathrm{d}}{\mathrm{d}x}(216x^{-2})$$

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}.

$$-\frac{4}{3}\times \left(216x^{-2}\right)^{-\frac{7}{3}}\left(-2\right)\times 216x^{-2-1}$$

Simplify.

$$576x^{-3}\times \left(216x^{-2}\right)^{-\frac{7}{3}}$$