Solve (+mathfrak{F}(2+1)(2^{2}+1)(2^4+1)(2^8+1)(2^16+1)(2^32+1)+1

$$( + \mathfrak { F } ( 2 + 1 ) ( 2 ^ { 2 } + 1 ) ( 2 ^ { 4 } + 1 ) ( 2 ^ { 8 } + 1 ) ( 2 ^ { 16 } + 1 ) ( 2 ^ { 32 } + 1 ) + 1$$

Evaluate

$$18446744073709551615F+1$$

Solution Steps

Add 2 and 1 to get 3.

$$F\times 3\left(2^{2}+1\right)\left(2^{4}+1\right)\left(2^{8}+1\right)\left(2^{16}+1\right)\left(2^{32}+1\right)+1$$

Calculate 2 to the power of 2 and get 4.

$$F\times 3\left(4+1\right)\left(2^{4}+1\right)\left(2^{8}+1\right)\left(2^{16}+1\right)\left(2^{32}+1\right)+1$$

Add 4 and 1 to get 5.

$$F\times 3\times 5\left(2^{4}+1\right)\left(2^{8}+1\right)\left(2^{16}+1\right)\left(2^{32}+1\right)+1$$

Multiply 3 and 5 to get 15.

$$F\times 15\left(2^{4}+1\right)\left(2^{8}+1\right)\left(2^{16}+1\right)\left(2^{32}+1\right)+1$$

Calculate 2 to the power of 4 and get 16.

$$F\times 15\left(16+1\right)\left(2^{8}+1\right)\left(2^{16}+1\right)\left(2^{32}+1\right)+1$$

Add 16 and 1 to get 17.

$$F\times 15\times 17\left(2^{8}+1\right)\left(2^{16}+1\right)\left(2^{32}+1\right)+1$$

Multiply 15 and 17 to get 255.

$$F\times 255\left(2^{8}+1\right)\left(2^{16}+1\right)\left(2^{32}+1\right)+1$$

Calculate 2 to the power of 8 and get 256.

$$F\times 255\left(256+1\right)\left(2^{16}+1\right)\left(2^{32}+1\right)+1$$

Add 256 and 1 to get 257.

$$F\times 255\times 257\left(2^{16}+1\right)\left(2^{32}+1\right)+1$$

Multiply 255 and 257 to get 65535.

$$F\times 65535\left(2^{16}+1\right)\left(2^{32}+1\right)+1$$

Calculate 2 to the power of 16 and get 65536.

$$F\times 65535\left(65536+1\right)\left(2^{32}+1\right)+1$$

Add 65536 and 1 to get 65537.

$$F\times 65535\times 65537\left(2^{32}+1\right)+1$$

Multiply 65535 and 65537 to get 4294967295.

$$F\times 4294967295\left(2^{32}+1\right)+1$$

Calculate 2 to the power of 32 and get 4294967296.

$$F\times 4294967295\left(4294967296+1\right)+1$$

Add 4294967296 and 1 to get 4294967297.

$$F\times 4294967295\times 4294967297+1$$

Multiply 4294967295 and 4294967297 to get 18446744073709551615.

$$F\times 18446744073709551615+1$$

Differentiate w.r.t. F

$$18446744073709551615$$

Steps Using Derivative Rule for Sum

Add 2 and 1 to get 3.

$$\frac{\mathrm{d}}{\mathrm{d}F}(F\times 3\left(2^{2}+1\right)\left(2^{4}+1\right)\left(2^{8}+1\right)\left(2^{16}+1\right)\left(2^{32}+1\right)+1)$$

Calculate 2 to the power of 2 and get 4.

$$\frac{\mathrm{d}}{\mathrm{d}F}(F\times 3\left(4+1\right)\left(2^{4}+1\right)\left(2^{8}+1\right)\left(2^{16}+1\right)\left(2^{32}+1\right)+1)$$

Add 4 and 1 to get 5.

$$\frac{\mathrm{d}}{\mathrm{d}F}(F\times 3\times 5\left(2^{4}+1\right)\left(2^{8}+1\right)\left(2^{16}+1\right)\left(2^{32}+1\right)+1)$$

Multiply 3 and 5 to get 15.

$$\frac{\mathrm{d}}{\mathrm{d}F}(F\times 15\left(2^{4}+1\right)\left(2^{8}+1\right)\left(2^{16}+1\right)\left(2^{32}+1\right)+1)$$

Calculate 2 to the power of 4 and get 16.

$$\frac{\mathrm{d}}{\mathrm{d}F}(F\times 15\left(16+1\right)\left(2^{8}+1\right)\left(2^{16}+1\right)\left(2^{32}+1\right)+1)$$

Add 16 and 1 to get 17.

$$\frac{\mathrm{d}}{\mathrm{d}F}(F\times 15\times 17\left(2^{8}+1\right)\left(2^{16}+1\right)\left(2^{32}+1\right)+1)$$

Multiply 15 and 17 to get 255.

$$\frac{\mathrm{d}}{\mathrm{d}F}(F\times 255\left(2^{8}+1\right)\left(2^{16}+1\right)\left(2^{32}+1\right)+1)$$

Calculate 2 to the power of 8 and get 256.

$$\frac{\mathrm{d}}{\mathrm{d}F}(F\times 255\left(256+1\right)\left(2^{16}+1\right)\left(2^{32}+1\right)+1)$$

Add 256 and 1 to get 257.

$$\frac{\mathrm{d}}{\mathrm{d}F}(F\times 255\times 257\left(2^{16}+1\right)\left(2^{32}+1\right)+1)$$

Multiply 255 and 257 to get 65535.

$$\frac{\mathrm{d}}{\mathrm{d}F}(F\times 65535\left(2^{16}+1\right)\left(2^{32}+1\right)+1)$$

Calculate 2 to the power of 16 and get 65536.

$$\frac{\mathrm{d}}{\mathrm{d}F}(F\times 65535\left(65536+1\right)\left(2^{32}+1\right)+1)$$

Add 65536 and 1 to get 65537.

$$\frac{\mathrm{d}}{\mathrm{d}F}(F\times 65535\times 65537\left(2^{32}+1\right)+1)$$

Multiply 65535 and 65537 to get 4294967295.

$$\frac{\mathrm{d}}{\mathrm{d}F}(F\times 4294967295\left(2^{32}+1\right)+1)$$

Calculate 2 to the power of 32 and get 4294967296.

$$\frac{\mathrm{d}}{\mathrm{d}F}(F\times 4294967295\left(4294967296+1\right)+1)$$

Add 4294967296 and 1 to get 4294967297.

$$\frac{\mathrm{d}}{\mathrm{d}F}(F\times 4294967295\times 4294967297+1)$$

Multiply 4294967295 and 4294967297 to get 18446744073709551615.

$$\frac{\mathrm{d}}{\mathrm{d}F}(F\times 18446744073709551615+1)$$

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}.

$$18446744073709551615F^{1-1}$$

Subtract 1 from 1.

$$18446744073709551615F^{0}$$

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

$$18446744073709551615\times 1$$

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

$$18446744073709551615$$