Solve 3y^{-1}v^-2*2v^-7y*2x^-9x^9

$$3 y ^ { - 1 } v ^ { - 2 } \cdot 2 v ^ { - 7 } y \cdot 2 x ^ { - 9 } x ^ { 9 }$$

Evaluate

$$\frac{12}{v^{9}}$$

Solution Steps

Multiply y^{-1} and y to get 1.

$$3v^{-2}\times 2v^{-7}\times 2x^{-9}x^{9}$$

To multiply powers of the same base, add their exponents. Add -2 and -7 to get -9.

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

Multiply x^{-9} and x^{9} to get 1.

$$3v^{-9}\times 2\times 2$$

Multiply 3 and 2 to get 6.

$$6v^{-9}\times 2$$

Multiply 6 and 2 to get 12.

$$12v^{-9}$$

Differentiate w.r.t. v

$$-\frac{108}{v^{10}}$$

Steps Using Definition of a Derivative

Multiply y^{-1} and y to get 1.

$$\frac{\mathrm{d}}{\mathrm{d}v}(3v^{-2}\times 2v^{-7}\times 2x^{-9}x^{9})$$

To multiply powers of the same base, add their exponents. Add -2 and -7 to get -9.

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

Multiply x^{-9} and x^{9} to get 1.

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

Multiply 3 and 2 to get 6.

$$\frac{\mathrm{d}}{\mathrm{d}v}(6v^{-9}\times 2)$$

Multiply 6 and 2 to get 12.

$$\frac{\mathrm{d}}{\mathrm{d}v}(12v^{-9})$$

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

$$-9\times 12v^{-9-1}$$

Multiply -9 times 12.

$$-108v^{-9-1}$$

Subtract 1 from -9.

$$-108v^{-10}$$