Solve (1/k^2)^4divk

$$( \frac { 1 } { k ^ { 2 } } ) ^ { 4 } \div k$$

Evaluate

$$\frac{1}{k^{9}}$$

Solution Steps

To raise \frac{1}{k^{2}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\frac{1^{4}}{\left(k^{2}\right)^{4}}}{k}$$

Express \frac{\frac{1^{4}}{\left(k^{2}\right)^{4}}}{k} as a single fraction.

$$\frac{1^{4}}{\left(k^{2}\right)^{4}k}$$

Calculate 1 to the power of 4 and get 1.

$$\frac{1}{\left(k^{2}\right)^{4}k}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 4 to get 8.

$$\frac{1}{k^{8}k}$$

To multiply powers of the same base, add their exponents. Add 8 and 1 to get 9.

$$\frac{1}{k^{9}}$$

Differentiate w.r.t. k

$$-\frac{9}{k^{10}}$$

Steps Using Chain Rule

To raise \frac{1}{k^{2}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\mathrm{d}}{\mathrm{d}k}(\frac{\frac{1^{4}}{\left(k^{2}\right)^{4}}}{k})$$

Express \frac{\frac{1^{4}}{\left(k^{2}\right)^{4}}}{k} as a single fraction.

$$\frac{\mathrm{d}}{\mathrm{d}k}(\frac{1^{4}}{\left(k^{2}\right)^{4}k})$$

Calculate 1 to the power of 4 and get 1.

$$\frac{\mathrm{d}}{\mathrm{d}k}(\frac{1}{\left(k^{2}\right)^{4}k})$$

To raise a power to another power, multiply the exponents. Multiply 2 and 4 to get 8.

$$\frac{\mathrm{d}}{\mathrm{d}k}(\frac{1}{k^{8}k})$$

To multiply powers of the same base, add their exponents. Add 8 and 1 to get 9.

$$\frac{\mathrm{d}}{\mathrm{d}k}(\frac{1}{k^{9}})$$

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).

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

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}.

$$-\left(k^{9}\right)^{-2}\times 9k^{9-1}$$

Simplify.

$$-9k^{8}\left(k^{9}\right)^{-2}$$