Solve n^-6/42*n^-7*n^7

$$\frac { n ^ { - 6 } } { 42 \cdot n ^ { - 7 } \cdot n ^ { 7 } }$$

Differentiate w.r.t. n

$$-\frac{1}{7n^{7}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}n}(\frac{1}{42n^{7}}n^{-6-\left(-7\right)})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}n}(\frac{1}{42n^{7}}n^{1})$$

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}.

$$\frac{1}{42n^{7}}n^{1-1}$$

Do the arithmetic.

$$\frac{1}{42n^{7}}n^{0}$$

For any term t except 0, t^{0}=1.

$$\frac{1}{42n^{7}}\times 1$$

For any term t, t\times 1=t and 1t=t.

$$\frac{1}{42n^{7}}$$

Evaluate

$$\frac{1}{42n^{6}}$$

Solution Steps

Multiply n^{-7} and n^{7} to get 1.

$$\frac{n^{-6}}{42}$$