**Calculus:** **Calculus** is one of the basic concepts of mathematics that deals with the continuous change of a function and finding the properties of derivatives and integrals of functions. Important Derivative Rules will help a lot for this.

**Derivatives:** The set of properties of calculus are derivatives. The derivative describes the instantaneous change of a function at any point with respect to one of its variable.

Derivative defines the slope of a function at any point.

Let’s consider y = f(x) is a function then, the derivative of y can be represented by ππ¦ππ₯ or yβ or fβ or ππππ₯.

**See Also: ****What is a derivative? Everything About it in One Post!**

**Top Important Derivative Rules to Simplify CalculusΒ **

Important derivative rules for computing the derivative of a function in calculus are presented here.

**Rule 1: Constant Rule**

The derivative of a constant is zero.

For any number c, f(x) = c then fβ (x) = 0

**Example:** Differentiate f(x)= 3. 3 is a constant and f(x)=3 is a horizontal line with a slope of zero. Therefore, its derivative is zero.

Hence fβ(x)= 0.

**Rule 2: Power Rule**

f(x) = π₯π then fβ(x) = nπ₯πβ1 βnβ can be a positive, negative, or a fraction. If the function is in the form of π₯π, we bring the power in front and reduce the current power by 1.

**Step1:** bringing the power in front.

**Step2:** reduce the current power by 1.

**Example:** Differentiate f(x)= π₯8.

**Step1:** Bringing the power in front (8π₯8 ).

**Step2:** reduce the current power by 1 (8π₯8β1). Therefore, fβ(x)= 8π₯7

**See Also: ****How To Find Derivative Of A Graph | Easiest Method**

**Rule 3: Constant Multiple Rule**

g(x) = c Β· f(x) then gβ(x) = c Β· fβ (x) If the function that is being differentiated is multiplied with a constant that makes no difference. It does not affect the process of differentiation. We ignore the constant until the final step and differentiate the rest of the function with the appropriate differential rule.

In the final step, we multiply the answer with the constant.

**Example:** Differentiate f(x)= 2π₯3

**Step 1:** Ignore the constant and differentiate the function. ο° (2) (3π₯2) [From Rule 2]

**Step 2:** Multiply the constant to the resultant derivative.

Therefore fβ(x) = 6π₯2

**Rule 4: Sum or Difference Rule. **

If f(x) = h(x) Β± g(x) then, ππππ₯ = πβππ₯ Β± ππππ₯

Derivative of a sum or difference of terms results in taking the derivative of each term separately.

**Example:**

Differentiate f(x) = 3π₯2 β 7x.

Here, let us consider k(x) = 3π₯2 ,

g(x) = 7x

=> ππππ₯ = 6x and ππππ₯ = 7.

Therefore, fβ(x) = 6x β 7.

These are some Basic Important Derivative Rules.

**Rule 5: Product Rule**

If f(x) = u(x)v(x) then,

fβ(x) = uβ(x)v(x) + u(x)vβ(x)

where, h(x), u(x), v(x) are there functions and uβ(x)= ππ’ππ₯ , vβ(x)= ππ£ππ₯

**Example:**

Differentiate f(x) = (6π₯2+ 2x) ( π₯3+ 1).

Let u(x) = 6π₯2 + 2x

v(x) = π₯3 + 1.

ο° ππ’ππ₯ = 12x + 2 and ππ£ππ₯ = 3π₯2 .

By using formula,

ππππ₯= u Γ ππ£ππ₯ + v Γ ππ’ππ₯

Therefore, ππππ₯ = (6π₯2 + 2x) (3π₯2) + (π₯3 + 1) (12x + 2)

= 18π₯4+ 6π₯3 + 12π₯4 + 2π₯3 + 12x + 2

= 30π₯4 + 8π₯3 + 12x + 2.

**See Also: ****What is a Partial Derivative? Scope and Application**

**Rule 6: Quotient Rule**

If f(x) =π’(π₯)π£(π₯) , where u(x) and v(x) are both functions of x then,

ππππ₯ = v Γ ππ’ππ₯ β u Γ ππ£ππ₯π£2

**Example:**

Differentiate f(x) = π₯2 + 7 3x β 1

Let u(x) = π₯2 + 7

v(x) = 3xβ1.

ππ’ππ₯ = 2x and ππ£ππ₯= 3

By using the formula,

ππππ₯ = v Γ ππ’ππ₯ β u Γ ππ£ππ₯π£2

ππππ₯ = (3x β 1)(2x) β (π₯2 + 7)(3) (3π₯β1)2

= 6π₯2β 2x β 3π₯2β 21 (3π₯β1)2

= 3π₯2 β 2x β 21 (3π₯β1)2

**Rule 7: Chain Rule**

If y is a function of u, i.e. y = f(u), and u is a function of x, i.e. u = g(x) then, the derivative of y with respect to x is

ππ¦ππ₯ = ππ¦ππ’ Γ ππ’ππ₯

**Example:**

Differentiate y = (π₯2β 5)4

Let u = π₯2 β 5

y = π’4.

β ππ’ππ₯ = 2x and ππ¦ππ’ = 4π’3.

By using the chain rule formula,

ππ¦ππ₯ = ππ¦ππ’ Γ ππ’ππ₯

= 4π’3 Γ 2x

= 4(π₯2β 5)3 Γ 2x

8x(π₯2β 5)3

This is one of the most Important Derivative Rules.

**Rule 8: Exponential Derivative Rules**

- If f(x) = ππ₯ then, fβ(x) = ln(a) ππ₯
- f(x) =ππ₯ then, fβ(x) = ππ₯
- If f(x) = ππ(π₯) then, fβ(x) = ln(a) ππ(π₯) gβ (x)
- f(x) = ππ(π₯) then, fβ(x) = ππ(π₯)gβ(x)

**Rule 9: Trignometry Derivative rules**

- If f(x) = sin(x) then, fβ(x) = cos(x)
- When f(x) = cos(x) then, fβ(x) = β sin(x)
- If f(x) = tan(x) then, fβ(x) = π ππ2(x)
- When f(x) = sec(x) then, fβ(x) = sec(x) tan(x)
- If f(x) = cot(x) then, fβ(x) = β πππ ππ2(x)
- When f(x) = cosec(x) then, fβ(x) = β cosec(x) cot(x)

**Rule 10: Logarithm Derivative rules**

- If f(x) = ππππ(x) then, fβ(x) = 1ln(π)π₯
- f(x) = ln(x) then, fβ(x) = 1π₯
- When f(x) = ππππ(g(x)) then, fβ(x) = πβ²(π₯)ln(π)π(π₯)
- f(x) = ln(g(x)) then, fβ(x) = πβ²(π₯)π(π₯)

**Also Refer: ****How To Find Derivative Of A Function? [Simple Ways]**