Solve (1/3a^3)^-4

$$( \frac { 1 } { 3 a ^ { 3 } } ) ^ { - 4 }$$

Evaluate

$$81a^{12}$$

Solution Steps

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

$$\frac{1^{-4}}{\left(3a^{3}\right)^{-4}}$$

Calculate 1 to the power of -4 and get 1.

$$\frac{1}{\left(3a^{3}\right)^{-4}}$$

Expand \left(3a^{3}\right)^{-4}.

$$\frac{1}{3^{-4}\left(a^{3}\right)^{-4}}$$

To raise a power to another power, multiply the exponents. Multiply 3 and -4 to get -12.

$$\frac{1}{3^{-4}a^{-12}}$$

Calculate 3 to the power of -4 and get \frac{1}{81}.

$$\frac{1}{\frac{1}{81}a^{-12}}$$

Differentiate w.r.t. a

$$972a^{11}$$

Steps Using Chain Rule

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{1^{-4}}{\left(3a^{3}\right)^{-4}})$$

Calculate 1 to the power of -4 and get 1.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{1}{\left(3a^{3}\right)^{-4}})$$

Expand \left(3a^{3}\right)^{-4}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{1}{3^{-4}\left(a^{3}\right)^{-4}})$$

To raise a power to another power, multiply the exponents. Multiply 3 and -4 to get -12.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{1}{3^{-4}a^{-12}})$$

Calculate 3 to the power of -4 and get \frac{1}{81}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{1}{\frac{1}{81}a^{-12}})$$

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).

$$-\left(\frac{1}{81}a^{-12}\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}a}(\frac{1}{81}a^{-12})$$

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}.

$$-\left(\frac{1}{81}a^{-12}\right)^{-2}\left(-12\right)\times \frac{1}{81}a^{-12-1}$$

Simplify.

$$\frac{4}{27}a^{-13}\times \left(\frac{1}{81}a^{-12}\right)^{-2}$$