Solve 4n^-1/n

$$\frac { 4 n ^ { - 1 } } { n }$$

Differentiate w.r.t. n

$$-\frac{8}{n^{3}}$$

Steps Using Derivative Rule for Product

For any two differentiable functions, the derivative of the product of two functions is the first function times the derivative of the second plus the second function times the derivative of the first.

$$4\times \frac{1}{n}\frac{\mathrm{d}}{\mathrm{d}n}(\frac{1}{n})+\frac{1}{n}\frac{\mathrm{d}}{\mathrm{d}n}(4\times \frac{1}{n})$$

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}.

$$4\times \frac{1}{n}\left(-1\right)n^{-1-1}+\frac{1}{n}\left(-1\right)\times 4n^{-1-1}$$

Simplify.

$$4\times \frac{1}{n}\left(-1\right)n^{-2}+\frac{1}{n}\left(-4\right)n^{-2}$$

To multiply powers of the same base, add their exponents.

$$-4n^{-1-2}-4n^{-1-2}$$

Simplify.

$$-4n^{-3}-4n^{-3}$$

Combine like terms.

$$\left(-4-4\right)n^{-3}$$

Add -4 to -4.

$$-8n^{-3}$$

Solution Steps

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

$$\frac{\mathrm{d}}{\mathrm{d}n}(\frac{4}{1}n^{-1-1})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}n}(4n^{-2})$$

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}.

$$-2\times 4n^{-2-1}$$

Do the arithmetic.

$$-8n^{-3}$$

Evaluate

$$\frac{4}{n^{2}}$$

Solution Steps

To divide powers of the same base, subtract the numerator's exponent from the denominator's exponent.

$$\frac{4}{n^{2}}$$