Solve n^8/(n^6)^4

$$\frac { n ^ { 8 } } { ( n ^ { 6 } ) ^ { 4 } }$$

Evaluate

$$\frac{1}{n^{16}}$$

Solution Steps

To raise a power to another power, multiply the exponents. Multiply 6 and 4 to get 24.

$$\frac{n^{8}}{n^{24}}$$

Rewrite n^{24} as n^{8}n^{16}. Cancel out n^{8} in both numerator and denominator.

$$\frac{1}{n^{16}}$$

Differentiate w.r.t. n

$$-\frac{16}{n^{17}}$$

Steps Using Chain Rule

To raise a power to another power, multiply the exponents. Multiply 6 and 4 to get 24.

$$\frac{\mathrm{d}}{\mathrm{d}n}(\frac{n^{8}}{n^{24}})$$

Rewrite n^{24} as n^{8}n^{16}. Cancel out n^{8} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}n}(\frac{1}{n^{16}})$$

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).

$$-\left(n^{16}\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}n}(n^{16})$$

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}.

$$-\left(n^{16}\right)^{-2}\times 16n^{16-1}$$

Simplify.

$$-16n^{15}\left(n^{16}\right)^{-2}$$