Solve (k^2)^-2*k^3/2k^3

$$\frac { ( k ^ { 2 } ) ^ { - 2 } \cdot k ^ { 3 } } { 2 k ^ { 3 } }$$

Evaluate

$$\frac{1}{2k^{4}}$$

Solution Steps

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

$$\frac{k^{-4}k^{3}}{2k^{3}}$$

To multiply powers of the same base, add their exponents. Add -4 and 3 to get -1.

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

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

$$\frac{1}{2k^{4}}$$

Differentiate w.r.t. k

$$-\frac{2}{k^{5}}$$

Steps Using Chain Rule

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

$$\frac{\mathrm{d}}{\mathrm{d}k}(\frac{k^{-4}k^{3}}{2k^{3}})$$

To multiply powers of the same base, add their exponents. Add -4 and 3 to get -1.

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

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

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

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).

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

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}.

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

Simplify.

$$-8k^{3}\times \left(2k^{4}\right)^{-2}$$