Solve (1/2r^6)^-6

$$( \frac { 1 } { 2 r ^ { 6 } } ) ^ { - 6 }$$

Evaluate

$$64r^{36}$$

Solution Steps

To raise \frac{1}{2r^{6}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{1^{-6}}{\left(2r^{6}\right)^{-6}}$$

Calculate 1 to the power of -6 and get 1.

$$\frac{1}{\left(2r^{6}\right)^{-6}}$$

Expand \left(2r^{6}\right)^{-6}.

$$\frac{1}{2^{-6}\left(r^{6}\right)^{-6}}$$

To raise a power to another power, multiply the exponents. Multiply 6 and -6 to get -36.

$$\frac{1}{2^{-6}r^{-36}}$$

Calculate 2 to the power of -6 and get \frac{1}{64}.

$$\frac{1}{\frac{1}{64}r^{-36}}$$

Differentiate w.r.t. r

$$2304r^{35}$$

Steps Using Chain Rule

To raise \frac{1}{2r^{6}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\mathrm{d}}{\mathrm{d}r}(\frac{1^{-6}}{\left(2r^{6}\right)^{-6}})$$

Calculate 1 to the power of -6 and get 1.

$$\frac{\mathrm{d}}{\mathrm{d}r}(\frac{1}{\left(2r^{6}\right)^{-6}})$$

Expand \left(2r^{6}\right)^{-6}.

$$\frac{\mathrm{d}}{\mathrm{d}r}(\frac{1}{2^{-6}\left(r^{6}\right)^{-6}})$$

To raise a power to another power, multiply the exponents. Multiply 6 and -6 to get -36.

$$\frac{\mathrm{d}}{\mathrm{d}r}(\frac{1}{2^{-6}r^{-36}})$$

Calculate 2 to the power of -6 and get \frac{1}{64}.

$$\frac{\mathrm{d}}{\mathrm{d}r}(\frac{1}{\frac{1}{64}r^{-36}})$$

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).

$$-\left(\frac{1}{64}r^{-36}\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}r}(\frac{1}{64}r^{-36})$$

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}.

$$-\left(\frac{1}{64}r^{-36}\right)^{-2}\left(-36\right)\times \frac{1}{64}r^{-36-1}$$

Simplify.

$$\frac{9}{16}r^{-37}\times \left(\frac{1}{64}r^{-36}\right)^{-2}$$