Chapter 3, Problem 1CRQ

(a)

To determine

To fill: The derivative of a constant term.

Explanation of Solution

If c is a constant, then $\frac{d}{dx}\left(c\right)$ is 0.

Assume $f\left(x\right)=c$ then use the first principal of derivative.

$\begin{array}{c}{f}^{\prime }\left(x\right)=\underset{h\to \infty }{\lim }\frac{f\left(x+h\right)-f\left(x\right)}{h}\\ =\underset{h\to \infty }{\lim }\frac{c-c}{h}\\ =\underset{h\to \infty }{\lim }0\\ =0\end{array}$

Therefore, the derivative of a constant function is equal to 0.

(b)

To determine

To fill: The power rule of derivative.

Explanation of Solution

The Power Rule states that if n is any real number, then $\frac{d}{dx}\left(x^n\right)$ is $n{x}^{n-1}$.

By the definition of Power Rule,

If n is any real number, then $\frac{d}{dx}\left(x^n\right)=n{x}^{n-1}$

Therefore, the derivative of a given expression $\frac{d}{dx}\left(x^n\right)$ is $n{x}^{n-1}$.

(c)

To determine

To fill: The constant multiple rule of derivative.

Explanation of Solution

The Constant Multiple Rule states that if c is a constant, then $\frac{d}{dx}\left[cf\left(x\right)\right]$ is $c\frac{d}{dx}\left[f\left(x\right)\right]$.

By the definition of Constant Multiple Rule,

The derivative of a constant times a differentiable function is equal to the constant times the derivative of the function. i.e. $\frac{d}{dx}\left[cf\left(x\right)\right]=c\frac{d}{dx}\left[f\left(x\right)\right]$.

Therefore, the derivative of a given expression $\frac{d}{dx}\left[cf\left(x\right)\right]$ is $c\frac{d}{dx}\left[f\left(x\right)\right]$.

(d)

To determine

To fill: The sum rule for two functions $f\left(x\right)$ and $g\left(x\right)$.

Explanation of Solution

The Sum Rule states that $\frac{d}{dx}\left[f\left(x\right)\pm g\left(x\right)\right]$ is $\frac{d}{dx}\left[f\left(x\right)\right]\pm \frac{d}{dx}\left[g\left(x\right)\right]$.

By the definition of Sum Rule,

The derivative of the sum(difference) of differentiable functions is equal to the sum(difference) of their derivatives. i... $\frac{d}{dx}\left[f\left(x\right)\pm g\left(x\right)\right]=\frac{d}{dx}\left[f\left(x\right)\right]\pm \frac{d}{dx}\left[g\left(x\right)\right]$

Therefore, the derivative of a given expression $\frac{d}{dx}\left[f\left(x\right)\pm g\left(x\right)\right]$ is $\frac{d}{dx}\left[f\left(x\right)\right]\pm \frac{d}{dx}\left[g\left(x\right)\right]$.