Solve (32x^15)^-frac{2{5}}

$$( 32 x ^ { 15 } ) ^ { - \frac { 2 } { 5 } }$$

Evaluate

$$\frac{1}{4x^{6}}$$

Solution Steps

Expand \left(32x^{15}\right)^{-\frac{2}{5}}.

$$32^{-\frac{2}{5}}\left(x^{15}\right)^{-\frac{2}{5}}$$

To raise a power to another power, multiply the exponents. Multiply 15 and -\frac{2}{5} to get -6.

$$32^{-\frac{2}{5}}x^{-6}$$

Calculate 32 to the power of -\frac{2}{5} and get \frac{1}{4}.

$$\frac{1}{4}x^{-6}$$

Differentiate w.r.t. x

$$-\frac{3}{2x^{7}}$$

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).

$$-\frac{2}{5}\times \left(32x^{15}\right)^{-\frac{2}{5}-1}\frac{\mathrm{d}}{\mathrm{d}x}(32x^{15})$$

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}.

$$-\frac{2}{5}\times \left(32x^{15}\right)^{-\frac{7}{5}}\times 15\times 32x^{15-1}$$

Simplify.

$$-192x^{14}\times \left(32x^{15}\right)^{-\frac{7}{5}}$$