Solve -(1/3k)*(-1/3k)

$$- ( \frac { 1 } { 3 k } ) \cdot ( - \frac { 1 } { 3 k } )$$

Evaluate

$$\frac{1}{9k^{2}}$$

Solution Steps

Multiply -\frac{1}{3k} and -\frac{1}{3k} to get \left(-\frac{1}{3k}\right)^{2}.

$$\left(-\frac{1}{3k}\right)^{2}$$

Calculate -\frac{1}{3k} to the power of 2 and get \left(\frac{1}{3k}\right)^{2}.

$$\left(\frac{1}{3k}\right)^{2}$$

To raise \frac{1}{3k} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{1^{2}}{\left(3k\right)^{2}}$$

Calculate 1 to the power of 2 and get 1.

$$\frac{1}{\left(3k\right)^{2}}$$

Expand \left(3k\right)^{2}.

$$\frac{1}{3^{2}k^{2}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{1}{9k^{2}}$$

Differentiate w.r.t. k

$$-\frac{2}{9k^{3}}$$

Steps Using Chain Rule

Multiply -\frac{1}{3k} and -\frac{1}{3k} to get \left(-\frac{1}{3k}\right)^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}k}(\left(-\frac{1}{3k}\right)^{2})$$

Calculate -\frac{1}{3k} to the power of 2 and get \left(\frac{1}{3k}\right)^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}k}(\left(\frac{1}{3k}\right)^{2})$$

To raise \frac{1}{3k} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\mathrm{d}}{\mathrm{d}k}(\frac{1^{2}}{\left(3k\right)^{2}})$$

Calculate 1 to the power of 2 and get 1.

$$\frac{\mathrm{d}}{\mathrm{d}k}(\frac{1}{\left(3k\right)^{2}})$$

Expand \left(3k\right)^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}k}(\frac{1}{3^{2}k^{2}})$$

Calculate 3 to the power of 2 and get 9.

$$\frac{\mathrm{d}}{\mathrm{d}k}(\frac{1}{9k^{2}})$$

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).

$$-\left(9k^{2}\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}k}(9k^{2})$$

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}.

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

Simplify.

$$-18k^{1}\times \left(9k^{2}\right)^{-2}$$

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

$$-18k\times \left(9k^{2}\right)^{-2}$$