Solve 6y^3-5y^2-7+5y+8-5y^3+7y^2=

$$6 y ^ { 3 } - 5 y ^ { 2 } - 7 + 5 y + 8 - 5 y ^ { 3 } + 7 y ^ { 2 } =$$

Evaluate

$$y^{3}+2y^{2}+5y+1$$

Solution Steps

Add -7 and 8 to get 1.

$$6y^{3}-5y^{2}+1+5y-5y^{3}+7y^{2}$$

Combine 6y^{3} and -5y^{3} to get y^{3}.

$$y^{3}-5y^{2}+1+5y+7y^{2}$$

Combine -5y^{2} and 7y^{2} to get 2y^{2}.

$$y^{3}+2y^{2}+1+5y$$

Differentiate w.r.t. y

$$3y^{2}+4y+5$$

Steps Using Derivative Rule for Sum

Add -7 and 8 to get 1.

$$\frac{\mathrm{d}}{\mathrm{d}y}(6y^{3}-5y^{2}+1+5y-5y^{3}+7y^{2})$$

Combine 6y^{3} and -5y^{3} to get y^{3}.

$$\frac{\mathrm{d}}{\mathrm{d}y}(y^{3}-5y^{2}+1+5y+7y^{2})$$

Combine -5y^{2} and 7y^{2} to get 2y^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}y}(y^{3}+2y^{2}+1+5y)$$

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}.

$$3y^{3-1}+2\times 2y^{2-1}+5y^{1-1}$$

Subtract 1 from 3.

$$3y^{2}+2\times 2y^{2-1}+5y^{1-1}$$

Multiply 2 times 2.

$$3y^{2}+4y^{2-1}+5y^{1-1}$$

Subtract 1 from 2.

$$3y^{2}+4y^{1}+5y^{1-1}$$

Subtract 1 from 1.

$$3y^{2}+4y^{1}+5y^{0}$$

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

$$3y^{2}+4y+5y^{0}$$

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

$$3y^{2}+4y+5\times 1$$

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

$$3y^{2}+4y+5$$