Solve {c}{x^{2}+x^{4}+5^5}{+6^6+7^7}{+8^8}

$$\left. \begin{array} { c } { x ^ { 2 } + x ^ { 4 } + 5 ^ { 5 } } \\ { + 6 ^ { 6 } + 7 ^ { 7 } } \\ { + 8 ^ { 8 } } \end{array} \right.$$

Evaluate

$$x^{4}+x^{2}[phone]$$

Solution Steps

Calculate 5 to the power of 5 and get 3125.

$$x^{2}+x^{4}+3125+6^{6}+7^{7}+8^{8}$$

Calculate 6 to the power of 6 and get 46656.

$$x^{2}+x^{4}+3125+46656+7^{7}+8^{8}$$

Add 3125 and 46656 to get 49781.

$$x^{2}+x^{4}+49781+7^{7}+8^{8}$$

Calculate 7 to the power of 7 and get 823543.

$$x^{2}+x^{4}+49781+823543+8^{8}$$

Add 49781 and 823543 to get 873324.

$$x^{2}+x^{4}+873324+8^{8}$$

Calculate 8 to the power of 8 and get 16777216.

$$x^{2}+x^{4}+873324+16777216$$

Add 873324 and 16777216 to get 17650540.

$$x^{2}+x^{4}[phone]$$

Differentiate w.r.t. x

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

Steps Using Derivative Rule for Sum

Calculate 5 to the power of 5 and get 3125.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}+x^{4}+3125+6^{6}+7^{7}+8^{8})$$

Calculate 6 to the power of 6 and get 46656.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}+x^{4}+3125+46656+7^{7}+8^{8})$$

Add 3125 and 46656 to get 49781.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}+x^{4}+49781+7^{7}+8^{8})$$

Calculate 7 to the power of 7 and get 823543.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}+x^{4}+49781+823543+8^{8})$$

Add 49781 and 823543 to get 873324.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}+x^{4}+873324+8^{8})$$

Calculate 8 to the power of 8 and get 16777216.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}+x^{4}+873324+16777216)$$

Add 873324 and 16777216 to get 17650540.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}+x^{4}[phone])$$

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}.

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

Subtract 1 from 2.

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

Subtract 1 from 4.

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

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

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