Solve 10x^4+7x+5x^5--7x^5+14-2x

$$10 { x }^{ 4 } +7x+5 { x }^{ 5 } --7 { x }^{ 5 } +14-2x$$

Evaluate

$$12x^{5}+10x^{4}+5x+14$$

Solution Steps

The opposite of -7x^{5} is 7x^{5}.

$$10x^{4}+7x+5x^{5}+7x^{5}+14-2x$$

Combine 5x^{5} and 7x^{5} to get 12x^{5}.

$$10x^{4}+7x+12x^{5}+14-2x$$

Combine 7x and -2x to get 5x.

$$10x^{4}+5x+12x^{5}+14$$

Differentiate w.r.t. x

$$60x^{4}+40x^{3}+5$$

Steps Using Derivative Rule for Sum

The opposite of -7x^{5} is 7x^{5}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(10x^{4}+7x+5x^{5}+7x^{5}+14-2x)$$

Combine 5x^{5} and 7x^{5} to get 12x^{5}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(10x^{4}+7x+12x^{5}+14-2x)$$

Combine 7x and -2x to get 5x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(10x^{4}+5x+12x^{5}+14)$$

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}.

$$4\times 10x^{4-1}+5x^{1-1}+5\times 12x^{5-1}$$

Multiply 4 times 10.

$$40x^{4-1}+5x^{1-1}+5\times 12x^{5-1}$$

Subtract 1 from 4.

$$40x^{3}+5x^{1-1}+5\times 12x^{5-1}$$

Subtract 1 from 1.

$$40x^{3}+5x^{0}+5\times 12x^{5-1}$$

Multiply 1 times 5.

$$40x^{3}+5x^{0}+60x^{5-1}$$

Subtract 1 from 5.

$$40x^{3}+5x^{0}+60x^{4}$$

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

$$40x^{3}+5\times 1+60x^{4}$$

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

$$40x^{3}+5+60x^{4}$$