Solve k^-36*k^9/k^6*k^0

$$\frac { k ^ { - 36 } \cdot k ^ { 9 } } { k ^ { 6 } \cdot k ^ { 0 } }$$

Evaluate

$$\frac{1}{k^{33}}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add -36 and 9 to get -27.

$$\frac{k^{-27}}{k^{6}k^{0}}$$

To multiply powers of the same base, add their exponents. Add 6 and 0 to get 6.

$$\frac{k^{-27}}{k^{6}}$$

Rewrite k^{6} as k^{-27}k^{33}. Cancel out k^{-27} in both numerator and denominator.

$$\frac{1}{k^{33}}$$

Differentiate w.r.t. k

$$-\frac{33}{k^{34}}$$

Steps Using Chain Rule

To multiply powers of the same base, add their exponents. Add -36 and 9 to get -27.

$$\frac{\mathrm{d}}{\mathrm{d}k}(\frac{k^{-27}}{k^{6}k^{0}})$$

To multiply powers of the same base, add their exponents. Add 6 and 0 to get 6.

$$\frac{\mathrm{d}}{\mathrm{d}k}(\frac{k^{-27}}{k^{6}})$$

Rewrite k^{6} as k^{-27}k^{33}. Cancel out k^{-27} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}k}(\frac{1}{k^{33}})$$

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).

$$-\left(k^{33}\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}k}(k^{33})$$

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}.

$$-\left(k^{33}\right)^{-2}\times 33k^{33-1}$$

Simplify.

$$-33k^{32}\left(k^{33}\right)^{-2}$$