Solve 3x^218x^3+x^2+4x-7

$$3 { x }^{ 2 } 18 { x }^{ 3 } + { x }^{ 2 } +4x-7$$

Evaluate

$$54x^{5}+x^{2}+4x-7$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 2 and 3 to get 5.

$$3x^{5}\times 18+x^{2}+4x-7$$

Multiply 3 and 18 to get 54.

$$54x^{5}+x^{2}+4x-7$$

Differentiate w.r.t. x

$$2\left(135x^{4}+x+2\right)$$

Steps Using Derivative Rule for Sum

To multiply powers of the same base, add their exponents. Add 2 and 3 to get 5.

$$\frac{\mathrm{d}}{\mathrm{d}x}(3x^{5}\times 18+x^{2}+4x-7)$$

Multiply 3 and 18 to get 54.

$$\frac{\mathrm{d}}{\mathrm{d}x}(54x^{5}+x^{2}+4x-7)$$

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}.

$$5\times 54x^{5-1}+2x^{2-1}+4x^{1-1}$$

Multiply 5 times 54.

$$270x^{5-1}+2x^{2-1}+4x^{1-1}$$

Subtract 1 from 5.

$$270x^{4}+2x^{2-1}+4x^{1-1}$$

Subtract 1 from 2.

$$270x^{4}+2x^{1}+4x^{1-1}$$

Subtract 1 from 1.

$$270x^{4}+2x^{1}+4x^{0}$$

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

$$270x^{4}+2x+4x^{0}$$

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

$$270x^{4}+2x+4\times 1$$

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

$$270x^{4}+2x+4$$