Solve 3^-2*36*x^-2div6*2^3*x^-9

$${ 3 }^{ -2 } \times 36 \times { x }^{ -2 } \div 6 \times { 2 }^{ 3 } \times { x }^{ -9 }$$

Evaluate

$$\frac{16}{3x^{11}}$$

Solution Steps

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

$$\frac{\frac{1}{9}\times 36x^{-2}}{6}\times 2^{3}x^{-9}$$

Multiply \frac{1}{9} and 36 to get 4.

$$\frac{4x^{-2}}{6}\times 2^{3}x^{-9}$$

Divide 4x^{-2} by 6 to get \frac{2}{3}x^{-2}.

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

Calculate 2 to the power of 3 and get 8.

$$\frac{2}{3}x^{-2}\times 8x^{-9}$$

Multiply \frac{2}{3} and 8 to get \frac{16}{3}.

$$\frac{16}{3}x^{-2}x^{-9}$$

To multiply powers of the same base, add their exponents. Add -2 and -9 to get -11.

$$\frac{16}{3}x^{-11}$$

Differentiate w.r.t. x

$$-\frac{176}{3x^{12}}$$

Steps Using Definition of a Derivative

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\frac{1}{9}\times 36x^{-2}}{6}\times 2^{3}x^{-9})$$

Multiply \frac{1}{9} and 36 to get 4.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{4x^{-2}}{6}\times 2^{3}x^{-9})$$

Divide 4x^{-2} by 6 to get \frac{2}{3}x^{-2}.

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

Calculate 2 to the power of 3 and get 8.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2}{3}x^{-2}\times 8x^{-9})$$

Multiply \frac{2}{3} and 8 to get \frac{16}{3}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{16}{3}x^{-2}x^{-9})$$

To multiply powers of the same base, add their exponents. Add -2 and -9 to get -11.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{16}{3}x^{-11})$$

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

$$-11\times \frac{16}{3}x^{-11-1}$$

Multiply -11 times \frac{16}{3}.

$$-\frac{176}{3}x^{-11-1}$$

Subtract 1 from -11.

$$-\frac{176}{3}x^{-12}$$