Solve left(3xright)^-2^left(3xright)^-2

$${ \left(3x \right) }^{ - { 2 }^{ } } { \left(3x \right) }^{ -2 }$$

Evaluate

$$\frac{1}{81x^{4}}$$

Solution Steps

Calculate 2 to the power of 1 and get 2.

$$\left(3x\right)^{-2}\times \left(3x\right)^{-2}$$

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

$$3^{-2}x^{-2}\times \left(3x\right)^{-2}$$

Calculate 3 to the power of -2 and get \frac{1}{9}.

$$\frac{1}{9}x^{-2}\times \left(3x\right)^{-2}$$

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

$$\frac{1}{9}x^{-2}\times 3^{-2}x^{-2}$$

Calculate 3 to the power of -2 and get \frac{1}{9}.

$$\frac{1}{9}x^{-2}\times \frac{1}{9}x^{-2}$$

Multiply \frac{1}{9} and \frac{1}{9} to get \frac{1}{81}.

$$\frac{1}{81}x^{-2}x^{-2}$$

To multiply powers of the same base, add their exponents. Add -2 and -2 to get -4.

$$\frac{1}{81}x^{-4}$$

Differentiate w.r.t. x

$$-\frac{4}{81x^{5}}$$

Steps Using Definition of a Derivative

Calculate 2 to the power of 1 and get 2.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(3x\right)^{-2}\times \left(3x\right)^{-2})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(3^{-2}x^{-2}\times \left(3x\right)^{-2})$$

Calculate 3 to the power of -2 and get \frac{1}{9}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{9}x^{-2}\times \left(3x\right)^{-2})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{9}x^{-2}\times 3^{-2}x^{-2})$$

Calculate 3 to the power of -2 and get \frac{1}{9}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{9}x^{-2}\times \frac{1}{9}x^{-2})$$

Multiply \frac{1}{9} and \frac{1}{9} to get \frac{1}{81}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{81}x^{-2}x^{-2})$$

To multiply powers of the same base, add their exponents. Add -2 and -2 to get -4.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{81}x^{-4})$$

The derivative of ax^{n} is nax^{n-1}.

$$-4\times \frac{1}{81}x^{-4-1}$$

Multiply -4 times \frac{1}{81}.

$$-\frac{4}{81}x^{-4-1}$$

Subtract 1 from -4.

$$-\frac{4}{81}x^{-5}$$