Solve (n^5)^-9

$$( n ^ { 5 } ) ^ { - 9 }$$

Evaluate

$$\frac{1}{n^{45}}$$

Steps for Raising Power to Another Power

Use the rules of exponents to simplify the expression.

$$\left(n^{5}\right)^{-9}$$

To raise a power to another power, multiply the exponents.

$$n^{5\left(-9\right)}$$

Multiply 5 times -9.

$$\frac{1}{n^{45}}$$

Differentiate w.r.t. n

$$-\frac{45}{n^{46}}$$

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).

$$-9\left(n^{5}\right)^{-9-1}\frac{\mathrm{d}}{\mathrm{d}n}(n^{5})$$

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}.

$$-9\left(n^{5}\right)^{-10}\times 5n^{5-1}$$

Simplify.

$$-45n^{4}\left(n^{5}\right)^{-10}$$