Question

Sum and constant-multiple rules: State the sum and constant-multiple rules for (a) derivatives and (b) limits.

Short answer

(a) The sum rule for derivatives states that the derivative of a sum is equal to the sum of the derivatives.

Sum rule of derivatives:

$\frac{d}{dx}\left[f\right(x)+g(x\left)\right]=\frac{d}{dx}f\left(x\right)+\frac{d}{dx}g\left(x\right)$

The Constant multiple rule says the derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function.

Constant multiple rule:

$\frac{d}{dx}[k·f(x\left)\right]=k·\frac{d}{dx}f\left(x\right)$

(b) Sum law for limits states that the limit of the sum of two functions equals to the sum of the limits of two functions.

Sum rule of limit:

$\underset{x\to c}{\lim }\left[f\right(x)+g(x\left)\right]=\underset{x\to c}{\lim }f\left(x\right)+\underset{x\to c}{\lim }g\left(x\right)$

Constant multiple law for limits states that the limit of a constant multiple of a function equals the product of the constant with the limit of the function.

Constant multiple of limit:

$\underset{x\to c}{\lim }[k·f(x\left)\right]=k·\underset{x\to c}{\lim }f\left(x\right)$

Step-by-step solution

Step 1. Given Information:

We want to State the sum and constant-multiple rules for

• (a) derivatives
• (b) limits.

Part (a) Step 1. Solution of (a):

The sum rule for derivatives states that the derivative of a sum is equal to the sum of the derivatives.

Sum rule:

$\frac{d}{dx}\left[f\right(x)+g(x\left)\right]=\frac{d}{dx}f\left(x\right)+\frac{d}{dx}g\left(x\right)$

The Constant multiple rule says the derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function.

Constant multiple rule:

$\frac{d}{dx}[k·f(x\left)\right]=k·\frac{d}{dx}f\left(x\right)$

Part (b) Step 1. Solution:

Sum law for limits states that the limit of the sum of two functions equals to the sum of the limits of two functions.

Sum Law:

$\frac{d}{dx}\left[f\right(x)+g(x\left)\right]=\frac{d}{dx}f\left(x\right)+\frac{d}{dx}g\left(x\right)$

Constant multiple law for limits states that the limit of a constant multiple of a function equals the product of the constant with the limit of the function.

Constant multiple law:

$\underset{x\to c}{\lim }[k·f(x\left)\right]=k·\underset{x\to c}{\lim }f\left(x\right)$