Solve (9x^3-27x^2+5x-2)+(-13x+5x^3-77x^3+5)

$$( 9 x ^ { 3 } - 27 x ^ { 2 } + 5 x - 2 ) + ( - 13 x + 5 x ^ { 3 } - 77 x ^ { 3 } + 5 )$$

Evaluate

$$3-8x-27x^{2}-63x^{3}$$

Solution Steps

Combine 5x and -13x to get -8x.

$$9x^{3}-27x^{2}-8x-2+5x^{3}-77x^{3}+5$$

Combine 9x^{3} and 5x^{3} to get 14x^{3}.

$$14x^{3}-27x^{2}-8x-2-77x^{3}+5$$

Combine 14x^{3} and -77x^{3} to get -63x^{3}.

$$-63x^{3}-27x^{2}-8x-2+5$$

Add -2 and 5 to get 3.

$$-63x^{3}-27x^{2}-8x+3$$

Differentiate w.r.t. x

$$-189x^{2}-54x-8$$

Steps Using Derivative Rule for Sum

Combine 5x and -13x to get -8x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(9x^{3}-27x^{2}-8x-2+5x^{3}-77x^{3}+5)$$

Combine 9x^{3} and 5x^{3} to get 14x^{3}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(14x^{3}-27x^{2}-8x-2-77x^{3}+5)$$

Combine 14x^{3} and -77x^{3} to get -63x^{3}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(-63x^{3}-27x^{2}-8x-2+5)$$

Add -2 and 5 to get 3.

$$\frac{\mathrm{d}}{\mathrm{d}x}(-63x^{3}-27x^{2}-8x+3)$$

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}.

$$3\left(-63\right)x^{3-1}+2\left(-27\right)x^{2-1}-8x^{1-1}$$

Multiply 3 times -63.

$$-189x^{3-1}+2\left(-27\right)x^{2-1}-8x^{1-1}$$

Subtract 1 from 3.

$$-189x^{2}+2\left(-27\right)x^{2-1}-8x^{1-1}$$

Multiply 2 times -27.

$$-189x^{2}-54x^{2-1}-8x^{1-1}$$

Subtract 1 from 2.

$$-189x^{2}-54x^{1}-8x^{1-1}$$

Subtract 1 from 1.

$$-189x^{2}-54x^{1}-8x^{0}$$

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

$$-189x^{2}-54x-8x^{0}$$

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

$$-189x^{2}-54x-8$$