Solve (2x^5+x^4+1x^3*4x^2+2x+2)

$$( 2 x ^ { 5 } + x ^ { 4 } + 1 x ^ { 3 } \cdot 4 x ^ { 2 } + 2 x + 2 )$$

Evaluate

$$6x^{5}+x^{4}+2x+2$$

Solution Steps

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

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

Multiply 1 and 4 to get 4.

$$2x^{5}+x^{4}+4x^{5}+2x+2$$

Combine 2x^{5} and 4x^{5} to get 6x^{5}.

$$6x^{5}+x^{4}+2x+2$$

Differentiate w.r.t. x

$$30x^{4}+4x^{3}+2$$

Steps Using Derivative Rule for Sum

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(2x^{5}+x^{4}+1x^{5}\times 4+2x+2)$$

Multiply 1 and 4 to get 4.

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

Combine 2x^{5} and 4x^{5} to get 6x^{5}.

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

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}.

$$5\times 6x^{5-1}+4x^{4-1}+2x^{1-1}$$

Multiply 5 times 6.

$$30x^{5-1}+4x^{4-1}+2x^{1-1}$$

Subtract 1 from 5.

$$30x^{4}+4x^{4-1}+2x^{1-1}$$

Subtract 1 from 4.

$$30x^{4}+4x^{3}+2x^{1-1}$$

Subtract 1 from 1.

$$30x^{4}+4x^{3}+2x^{0}$$

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

$$30x^{4}+4x^{3}+2\times 1$$

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

$$30x^{4}+4x^{3}+2$$