Solve 15n^53n^545n^2

$$15 n ^ { 5 } 3 n ^ { 5 } 45 n ^ { 2 }$$

Evaluate

$$2025n^{12}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 5 and 5 to get 10.

$$15n^{10}\times 3\times 45n^{2}$$

To multiply powers of the same base, add their exponents. Add 10 and 2 to get 12.

$$15n^{12}\times 3\times 45$$

Multiply 15 and 3 to get 45.

$$45n^{12}\times 45$$

Multiply 45 and 45 to get 2025.

$$2025n^{12}$$

Differentiate w.r.t. n

$$24300n^{11}$$

Steps Using Definition of a Derivative

To multiply powers of the same base, add their exponents. Add 5 and 5 to get 10.

$$\frac{\mathrm{d}}{\mathrm{d}n}(15n^{10}\times 3\times 45n^{2})$$

To multiply powers of the same base, add their exponents. Add 10 and 2 to get 12.

$$\frac{\mathrm{d}}{\mathrm{d}n}(15n^{12}\times 3\times 45)$$

Multiply 15 and 3 to get 45.

$$\frac{\mathrm{d}}{\mathrm{d}n}(45n^{12}\times 45)$$

Multiply 45 and 45 to get 2025.

$$\frac{\mathrm{d}}{\mathrm{d}n}(2025n^{12})$$

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

$$12\times 2025n^{12-1}$$

Multiply 12 times 2025.

$$24300n^{12-1}$$

Subtract 1 from 12.

$$24300n^{11}$$