Solve 3x*2x^2*3x^3*5x^4

$$3 x \cdot 2 x ^ { 2 } \cdot 3 x ^ { 3 } \cdot 5 x ^ { 4 }$$

Evaluate

$$90x^{10}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

$$3x^{3}\times 2\times 3x^{3}\times 5x^{4}$$

To multiply powers of the same base, add their exponents. Add 3 and 3 to get 6.

$$3x^{6}\times 2\times 3\times 5x^{4}$$

To multiply powers of the same base, add their exponents. Add 6 and 4 to get 10.

$$3x^{10}\times 2\times 3\times 5$$

Multiply 3 and 2 to get 6.

$$6x^{10}\times 3\times 5$$

Multiply 6 and 3 to get 18.

$$18x^{10}\times 5$$

Multiply 18 and 5 to get 90.

$$90x^{10}$$

Differentiate w.r.t. x

$$900x^{9}$$

Steps Using Definition of a Derivative

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

$$\frac{\mathrm{d}}{\mathrm{d}x}(3x^{3}\times 2\times 3x^{3}\times 5x^{4})$$

To multiply powers of the same base, add their exponents. Add 3 and 3 to get 6.

$$\frac{\mathrm{d}}{\mathrm{d}x}(3x^{6}\times 2\times 3\times 5x^{4})$$

To multiply powers of the same base, add their exponents. Add 6 and 4 to get 10.

$$\frac{\mathrm{d}}{\mathrm{d}x}(3x^{10}\times 2\times 3\times 5)$$

Multiply 3 and 2 to get 6.

$$\frac{\mathrm{d}}{\mathrm{d}x}(6x^{10}\times 3\times 5)$$

Multiply 6 and 3 to get 18.

$$\frac{\mathrm{d}}{\mathrm{d}x}(18x^{10}\times 5)$$

Multiply 18 and 5 to get 90.

$$\frac{\mathrm{d}}{\mathrm{d}x}(90x^{10})$$

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

$$10\times 90x^{10-1}$$

Multiply 10 times 90.

$$900x^{10-1}$$

Subtract 1 from 10.

$$900x^{9}$$