Solve {l}{a+a^7}{-7+8-9}{-a^2+9}{-89-7}{+9}

$$\left. \begin{array} { l } { a + a ^ { 7 } } \\ { - 7 + 8 - 9 } \\ { - a ^ { 2 } + 9 } \\ { - 89 - 7 } \\ { + 9 } \end{array} \right.$$

Evaluate

$$a^{7}-a^{2}+a-86$$

Solution Steps

Add -7 and 8 to get 1.

$$a+a^{7}+1-9-a^{2}+9-89-7+9$$

Subtract 9 from 1 to get -8.

$$a+a^{7}-8-a^{2}+9-89-7+9$$

Add -8 and 9 to get 1.

$$a+a^{7}+1-a^{2}-89-7+9$$

Subtract 89 from 1 to get -88.

$$a+a^{7}-88-a^{2}-7+9$$

Subtract 7 from -88 to get -95.

$$a+a^{7}-95-a^{2}+9$$

Add -95 and 9 to get -86.

$$a+a^{7}-86-a^{2}$$

Differentiate w.r.t. a

$$7a^{6}-2a+1$$

Steps Using Derivative Rule for Sum

Add -7 and 8 to get 1.

$$\frac{\mathrm{d}}{\mathrm{d}a}(a+a^{7}+1-9-a^{2}+9-89-7+9)$$

Subtract 9 from 1 to get -8.

$$\frac{\mathrm{d}}{\mathrm{d}a}(a+a^{7}-8-a^{2}+9-89-7+9)$$

Add -8 and 9 to get 1.

$$\frac{\mathrm{d}}{\mathrm{d}a}(a+a^{7}+1-a^{2}-89-7+9)$$

Subtract 89 from 1 to get -88.

$$\frac{\mathrm{d}}{\mathrm{d}a}(a+a^{7}-88-a^{2}-7+9)$$

Subtract 7 from -88 to get -95.

$$\frac{\mathrm{d}}{\mathrm{d}a}(a+a^{7}-95-a^{2}+9)$$

Add -95 and 9 to get -86.

$$\frac{\mathrm{d}}{\mathrm{d}a}(a+a^{7}-86-a^{2})$$

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}.

$$a^{1-1}+7a^{7-1}+2\left(-1\right)a^{2-1}$$

Subtract 1 from 1.

$$a^{0}+7a^{7-1}+2\left(-1\right)a^{2-1}$$

Subtract 1 from 7.

$$a^{0}+7a^{6}+2\left(-1\right)a^{2-1}$$

Multiply 7 times 1.

$$a^{0}+7a^{6}-2a^{2-1}$$

Subtract 1 from 2.

$$a^{0}+7a^{6}-2a^{1}$$

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

$$a^{0}+7a^{6}-2a$$

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

$$1+7a^{6}-2a$$