Solve (2x^4)(-6x^3)+(9x)(3x^6)

$$( 2 x ^ { 4 } ) ( - 6 x ^ { 3 } ) + ( 9 x ) ( 3 x ^ { 6 } )$$

Evaluate

$$15x^{7}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 4 and 3 to get 7.

$$2x^{7}\left(-6\right)+9x\times 3x^{6}$$

To multiply powers of the same base, add their exponents. Add 1 and 6 to get 7.

$$2x^{7}\left(-6\right)+9x^{7}\times 3$$

Multiply 2 and -6 to get -12.

$$-12x^{7}+9x^{7}\times 3$$

Multiply 9 and 3 to get 27.

$$-12x^{7}+27x^{7}$$

Combine -12x^{7} and 27x^{7} to get 15x^{7}.

$$15x^{7}$$

Differentiate w.r.t. x

$$105x^{6}$$

Steps Using Derivative Rule for Sum

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$4\left(-12x^{3}\right)x^{4-1}+27x^{6}x^{1-1}$$

Multiply 4 times -6\times 2x^{3}.

$$\left(-48x^{3}\right)x^{4-1}+27x^{6}x^{1-1}$$

Subtract 1 from 4.

$$\left(-48x^{3}\right)x^{3}+27x^{6}x^{1-1}$$

Subtract 1 from 1.

$$\left(-48x^{3}\right)x^{3}+27x^{6}x^{0}$$

For any term t except 0, t^{0}=1.

$$\left(-48x^{3}\right)x^{3}+27x^{6}\times 1$$

For any term t, t\times 1=t and 1t=t.

$$\left(-48x^{3}\right)x^{3}+27x^{6}$$