# Top Important Derivative Rules to Simplify Your Calculus

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 ππ¦ππ₯ or yβ or fβ or ππππ₯.

## 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₯πβ1 βnβ 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₯8β1). Therefore, fβ(x)= 8π₯7

### 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, ππππ₯ = πβππ₯ Β± ππππ₯

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
=> ππππ₯ = 6x and ππππ₯ = 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)= ππ’ππ₯ , vβ(x)= ππ£ππ₯

Example:

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

Let u(x) = 6π₯2 + 2x
v(x) = π₯3 + 1.
ο° ππ’ππ₯ = 12x + 2 and ππ£ππ₯ = 3π₯2 .

By using formula,
ππππ₯= u Γ ππ£ππ₯ + v Γ ππ’ππ₯

Therefore, ππππ₯ = (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.

### Rule 6: Quotient Rule

If f(x) =π’(π₯)π£(π₯) , where u(x) and v(x) are both functions of x then,
ππππ₯ = v Γ ππ’ππ₯ β u Γ ππ£ππ₯π£2

Example:

Differentiate f(x) = π₯2 + 7 3x β 1

Let u(x) = π₯2 + 7
v(x) = 3xβ1.
ππ’ππ₯ = 2x and ππ£ππ₯= 3

By using the formula,
ππππ₯ = v Γ ππ’ππ₯ β u Γ ππ£ππ₯π£2
ππππ₯ = (3x β 1)(2x) β (π₯2 + 7)(3) (3π₯β1)2
= 6π₯2β 2x β 3π₯2β 21 (3π₯β1)2
= 3π₯2 β 2x β 21 (3π₯β1)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
ππ¦ππ₯ = ππ¦ππ’ Γ ππ’ππ₯

Example:

Differentiate y = (π₯2β 5)4

Let u = π₯2 β 5
y = π’4.
β ππ’ππ₯ = 2x and ππ¦ππ’ = 4π’3.

By using the chain rule formula,

ππ¦ππ₯ = ππ¦ππ’ Γ ππ’ππ₯

= 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) = ππ₯ then, fβ(x) = ln(a) ππ₯
• f(x) =ππ₯ then, fβ(x) = ππ₯
• If f(x) = ππ(π₯) then, fβ(x) = ln(a) ππ(π₯) gβ (x)
• f(x) = ππ(π₯) then, fβ(x) = ππ(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 ππ2(x)
• When f(x) = sec(x) then, fβ(x) = sec(x) tan(x)
• If f(x) = cot(x) then, fβ(x) = β πππ ππ2(x)
• When f(x) = cosec(x) then, fβ(x) = β cosec(x) cot(x)

### Rule 10: Logarithm Derivative rules

• If f(x) = ππππ(x) then, fβ(x) = 1ln(π)π₯
• f(x) = ln(x) then, fβ(x) = 1π₯
• When f(x) = ππππ(g(x)) then, fβ(x) = πβ²(π₯)ln(π)π(π₯)
• f(x) = ln(g(x)) then, fβ(x) = πβ²(π₯)π(π₯)