Solve n^0*n^-1*n^-78/n^32

$$\frac { n ^ { 0 } \cdot n ^ { - 1 } \cdot n ^ { - 78 } } { n ^ { 32 } }$$

Evaluate

$$\frac{1}{n^{111}}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 0 and -1 to get -1.

$$\frac{n^{-1}n^{-78}}{n^{32}}$$

To multiply powers of the same base, add their exponents. Add -1 and -78 to get -79.

$$\frac{n^{-79}}{n^{32}}$$

Rewrite n^{32} as n^{-79}n^{111}. Cancel out n^{-79} in both numerator and denominator.

$$\frac{1}{n^{111}}$$

Differentiate w.r.t. n

$$-\frac{111}{n^{112}}$$

Steps Using Chain Rule

To multiply powers of the same base, add their exponents. Add 0 and -1 to get -1.

$$\frac{\mathrm{d}}{\mathrm{d}n}(\frac{n^{-1}n^{-78}}{n^{32}})$$

To multiply powers of the same base, add their exponents. Add -1 and -78 to get -79.

$$\frac{\mathrm{d}}{\mathrm{d}n}(\frac{n^{-79}}{n^{32}})$$

Rewrite n^{32} as n^{-79}n^{111}. Cancel out n^{-79} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}n}(\frac{1}{n^{111}})$$

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).

$$-\left(n^{111}\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}n}(n^{111})$$

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}.

$$-\left(n^{111}\right)^{-2}\times 111n^{111-1}$$

Simplify.

$$-111n^{110}\left(n^{111}\right)^{-2}$$