Solve (32x^3)^-frac{6{5}}

$$( 32 x ^ { 3 } ) ^ { - \frac { 6 } { 5 } }$$

Evaluate

$$\frac{1}{64x^{\frac{18}{5}}}$$

Solution Steps

Expand \left(32x^{3}\right)^{-\frac{6}{5}}.

$$32^{-\frac{6}{5}}\left(x^{3}\right)^{-\frac{6}{5}}$$

To raise a power to another power, multiply the exponents. Multiply 3 and -\frac{6}{5} to get -\frac{18}{5}.

$$32^{-\frac{6}{5}}x^{-\frac{18}{5}}$$

Calculate 32 to the power of -\frac{6}{5} and get \frac{1}{64}.

$$\frac{1}{64}x^{-\frac{18}{5}}$$

Differentiate w.r.t. x

$$-\frac{9}{160x^{\frac{23}{5}}}$$

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).

$$-\frac{6}{5}\times \left(32x^{3}\right)^{-\frac{6}{5}-1}\frac{\mathrm{d}}{\mathrm{d}x}(32x^{3})$$

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}.

$$-\frac{6}{5}\times \left(32x^{3}\right)^{-\frac{11}{5}}\times 3\times 32x^{3-1}$$

Simplify.

$$-\frac{576}{5}x^{2}\times \left(32x^{3}\right)^{-\frac{11}{5}}$$