Solve f(0)+f(1)+f(2)+f(3)

$$f ( 0 ) + f ( 1 ) + f ( 2 ) + f ( 3 )$$

Evaluate

$$6f$$

Solution Steps

Anything times zero gives zero.

$$0+f\times 1+f\times 2+f\times 3$$

Combine f\times 1 and f\times 2 to get 3f.

$$0+3f+f\times 3$$

Combine 3f and f\times 3 to get 6f.

$$0+6f$$

Anything plus zero gives itself.

$$6f$$

Differentiate w.r.t. f

$$6$$

Steps Using Definition of a Derivative

Anything times zero gives zero.

$$\frac{\mathrm{d}}{\mathrm{d}f}(0+f\times 1+f\times 2+f\times 3)$$

Combine f\times 1 and f\times 2 to get 3f.

$$\frac{\mathrm{d}}{\mathrm{d}f}(0+3f+f\times 3)$$

Combine 3f and f\times 3 to get 6f.

$$\frac{\mathrm{d}}{\mathrm{d}f}(0+6f)$$

Anything plus zero gives itself.

$$\frac{\mathrm{d}}{\mathrm{d}f}(6f)$$

The derivative of ax^{n} is nax^{n-1}.

$$6f^{1-1}$$

Subtract 1 from 1.

$$6f^{0}$$

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

$$6\times 1$$

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

$$6$$