Solve f(x)=x^100+x^99+-----+x+1.text{then}f^prime(1)

$$f ( x ) = x ^ { 100 } + x ^ { 99 } + - - - - - + x + 1 . \text { then } f ^ { \prime } ( 1 )$$

Evaluate

$$x^{100}+x^{99}-x+1$$

Solution Steps

Multiply -1 and -1 to get 1.

$$x^{100}+x^{99}-\left(-\left(-x\right)\right)+1$$

Multiply -1 and -1 to get 1.

$$x^{100}+x^{99}-x+1$$

Differentiate w.r.t. x

$$100x^{99}+99x^{98}-1$$

Steps Using Derivative Rule for Sum

Multiply -1 and -1 to get 1.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{100}+x^{99}-\left(-\left(-x\right)\right)+1)$$

Multiply -1 and -1 to get 1.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{100}+x^{99}-x+1)$$

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}.

$$100x^{100-1}+99x^{99-1}-x^{1-1}$$

Subtract 1 from 100.

$$100x^{99}+99x^{99-1}-x^{1-1}$$

Subtract 1 from 99.

$$100x^{99}+99x^{98}-x^{1-1}$$

Subtract 1 from 1.

$$100x^{99}+99x^{98}-x^{0}$$

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

$$100x^{99}+99x^{98}-1$$