The h-derivative and the q-derivative of a function $\nu$ have been defined by the quotients respectively. The h-derivative is usually denoted by the quotient ${\mathcal{D}}_h\nu \left(\gamma \right)=\frac{d_h\nu \left(\gamma \right)}{d_h\gamma }$ while the q-derivative is denoted by the quotient ${\mathcal{D}}_q\nu \left(\gamma \right)=\frac{d_q\nu \left(\gamma \right)}{d_q\gamma }$, where $d_h\nu \left(\gamma \right)=\nu (\gamma +h)-\nu \left(\gamma \right)$ is called the h-differential and $d_q\nu \left(\gamma \right)=\nu \left(q\gamma \right)-\nu \left(\gamma \right)$ is called the q-differential for the function $\nu$. As an example the h-derivative and the q-derivative of ${\gamma }^n$ can be computed in the forms $\frac{{(\gamma +h)}^n-{\gamma }^n}{h}=n{\gamma }^{n-1}+\frac{n(n-1)}{2}{\gamma }^{n-2}h+...+h^{n-1}$ and $\frac{q^n-1}{q-1}{\gamma }^{n-1}=(q^{n-1}+...+1){\gamma }^{n-1}$ respectively. For the sake of simplicity, the notation ${\left[n\right]}_q$ is used instead of $\frac{q^n-1}{q-1}$, and then ${\mathcal{D}}_q{\gamma }^n={\left[n\right]}_q{\gamma }^{n-1}$. Since $\underset{q\to 1}{lim}{\mathcal{D}}_q\nu \left(\gamma \right)=\underset{h\to 0}{lim}{\mathcal{D}}_h\nu \left(\gamma \right)=\frac{d\nu \left(\gamma \right)}{d\gamma }$, the h-derivative and the q-derivative are generalizations of ordinary derivative. The q-derivative leads to the subject of q-calculus, see [5] for details.

The sum and product formula of q-derivatives for functions ${\nu }_1$ and ${\nu }_2$ are given by; and respectively. Since ${\nu }_1\left(\gamma \right){\nu }_2\left(\gamma \right)={\nu }_2\left(\gamma \right){\nu }_1\left(\gamma \right)$, (2) is equivalent to the upcoming formula

In view of (2), the quotient formula of q-derivatives is given by;

In view of (3), the quotient formula of q-derivatives is given by;

The formulae of h-derivatives are as follows: and

Next, we give the definition of q-derivative on a finite interval.

Function $\mu$ is called q-differentiable on $[a,b]$ if ${}_a{\mathcal{D}}_q\mu \left(\xi \right)$ exists for all $\xi \in [a,b]$. For $a=0$, we have ${}_0{\mathcal{D}}_q\mu \left(\xi \right)={\mathcal{D}}_q\mu \left(\xi \right)$ and ${\mathcal{D}}_q\mu \left(\xi \right)$ is the q-derivative of $\mu$ at $\xi \in [a,b]$ defined as follows:

The q-integral of function $\mu$ on interval $[a,b]$ is defined as follows:

In (11), by setting $a=0$, the Jackson q-definite integral given in [5], is deduced as follows:

If $c\in (a,\xi )$, then the q-definite integral on $[c,\xi ]$ is calculated as follows:

We are intent to unify the q-derivative and h-derivative in a single notion which will be named $q-h$-derivative. We give sum/difference, product and quotient formulas for $q-h$-derivatives, also the definition of $q-h$-integral is given. Further, we will define $q-h$-derivative and $q-h$-integral on finite interval. The composite derivatives and integrals will provide the opportunity to study theoretical and practical concepts and problems of different fields related to q-derivative and h-derivative simultaneously. This paper will be interesting and productive for scientists and engineers.

The $(q-h)$-differential of a real valued function $\mu$ is defined by;

Then for $h=0$, and $q\to 1$ in (14), we have and

In particular,

Then for $h=0$, and $q\to 1$ in (15), we have

For $u\left(\xi \right)=\mu \left(\xi \right)+\nu \left(\xi \right)$ the $(q-h)$-differential of u is given by;

For $\alpha \in \mathbb{R}$ the $(q-h)$-differential of $\alpha \mu$ is given by;

From (17) and (18) one can see that $(q-h)$-differential is linear. Here we see that if $p\left(\xi \right)=\mu \left(\xi \right)\nu \left(\xi \right)$, then $(q-h)$-differential is calculated as follows:

Hence we get

For $h=0$, and $q\to 1$ in (19), we have and respectively. Next, we define the $q-h$-derivative as follows:

For $h=0$ and $q\to 1$ in (20), we have and

By setting $h=0,q\to 1$ in (20), we get the ordinary derivative of $\mu$, provided the limit exists.

For $h=0$ and $q\to 1$ in (23), we have and

In particular we have $\underset{h\to 0}{lim}{\mathcal{C}}_h{\mathcal{D}}_1\left({\xi }^n\right)=n{\xi }^{n-1}$.

Here throughout the section, $I:=[a,b]$ for $a,b\in \mathbb{R}$. The $q-h-$derivative on I is given in the upcoming definition.

We say $\mu$ is left $q-h$-differentiable on $(a,x+h)$, if for each of its point ${\mathcal{C}}_h{\mathcal{D}}_q^{a^{+}}\mu \left(\xi \right)$ exists, and $\mu$ is called right $q-h$-differentiable on $(\xi +h,b)$, if at each of its point ${\mathcal{C}}_h{\mathcal{D}}_q^{b_{-}}\mu \left(\xi \right)$ exists. One can see that ${\mathcal{C}}_h{\mathcal{D}}_q^{a^{+}}\mu \left(b\right)={\mathcal{C}}_h{\mathcal{D}}_q^{b_{-}}\mu \left(a\right)$. In (38), by setting $h=0$ one can get the q-derivative defined in Definition 1, i.e. ${\mathcal{C}}_0{\mathcal{D}}_q^{a^{+}}\mu \left(\xi \right)={}_a{\mathcal{D}}_q\mu \left(\xi \right)$. Also for $a=0$ one can have ${\mathcal{C}}_h{\mathcal{D}}_q^{0^{+}}\mu \left(\xi \right)=C{}_h{\mathcal{D}}_q\mu \left(\xi \right)$, i.e. the $q-h$-derivative given in (20) is deduced; for $h=0=a$ one can have ${\mathcal{C}}_0{\mathcal{D}}_q^{0^{+}}\mu \left(\xi \right)={\mathcal{D}}_q\mu \left(\xi \right)$, i.e. the q-derivative is deduced; for $a=0,q=1$ one can have ${\mathcal{C}}_h{\mathcal{D}}_1^{0^{+}}\mu \left(\xi \right)={\mathcal{D}}_h\mu \left(\xi \right)$ i.e. the h-derivative is deduced; for $h=0=a$ and taking limit $q\to 1$ one can get the usual derivative for a differentiable function $\mu$ i.e. $\underset{q\to 1}{lim}{\mathcal{C}}_0{\mathcal{D}}_q^{0^{+}}\mu \left(\xi \right)=\frac{d}{d\xi }\mu \left(\xi \right)$. One can get similar results from equation (39). The definition of left and right $q-$derivatives defined on I can be obtained from (39) by setting $h=0$ as follows:

It is notable that from (40), we have ${\mathcal{D}}_q^{0^{+}}\mu \left(\xi \right)={\mathcal{D}}_q\mu \left(\xi \right)$ i.e. the left $q-$derivative coincides with $q-$derivative defined in Definition 1.