# 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 𝑑𝑦𝑑𝑥 or y’ or f’ or 𝑑𝑓𝑑𝑥.

## Top Important Derivative Rules to Simplify Calculus

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 f’ (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 f’(x)= 0.

### Rule 2: Power Rule

f(x) = 𝑥𝑛 then f’(x) = n𝑥𝑛−1 ‘n’ can be a positive, negative, or a fraction. If the function is in the form of 𝑥𝑛, 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)= 𝑥8.

Step1: Bringing the power in front (8𝑥8 ).

Step2: reduce the current power by 1 (8𝑥8−1). Therefore, f’(x)= 8𝑥7

### Rule 3: Constant Multiple Rule

g(x) = c · f(x) then g’(x) = c · f’ (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)= 2𝑥3

Step 1: Ignore the constant and differentiate the function.  (2) (3𝑥2) [From Rule 2]

Step 2: Multiply the constant to the resultant derivative.

Therefore f’(x) = 6𝑥2

### Rule 4: Sum or Difference Rule.

If f(x) = h(x) ± g(x) then, 𝑑𝑓𝑑𝑥 = 𝑑ℎ𝑑𝑥 ± 𝑑𝑔𝑑𝑥

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

Example:

Differentiate f(x) = 3𝑥2 − 7x.

Here, let us consider k(x) = 3𝑥2 ,
g(x) = 7x
=> 𝑑𝑘𝑑𝑥 = 6x and 𝑑𝑔𝑑𝑥 = 7.

Therefore, f’(x) = 6x − 7.

These are some Basic Important Derivative Rules.

### Rule 5: Product Rule

If f(x) = u(x)v(x) then,
f’(x) = u’(x)v(x) + u(x)v’(x)
where, h(x), u(x), v(x) are there functions and u’(x)= 𝑑𝑢𝑑𝑥 , v’(x)= 𝑑𝑣𝑑𝑥

Example:

Differentiate f(x) = (6𝑥2+ 2x) ( 𝑥3+ 1).

Let u(x) = 6𝑥2 + 2x
v(x) = 𝑥3 + 1.
 𝑑𝑢𝑑𝑥 = 12x + 2 and 𝑑𝑣𝑑𝑥 = 3𝑥2 .

By using formula,
𝑑𝑓𝑑𝑥= u × 𝑑𝑣𝑑𝑥 + v × 𝑑𝑢𝑑𝑥

Therefore, 𝑑𝑓𝑑𝑥 = (6𝑥2 + 2x) (3𝑥2) + (𝑥3 + 1) (12x + 2)
= 18𝑥4+ 6𝑥3 + 12𝑥4 + 2𝑥3 + 12x + 2
= 30𝑥4 + 8𝑥3 + 12x + 2.

### Rule 6: Quotient Rule

If f(x) =𝑢(𝑥)𝑣(𝑥) , where u(x) and v(x) are both functions of x then,
𝑑𝑓𝑑𝑥 = v × 𝑑𝑢𝑑𝑥 − u × 𝑑𝑣𝑑𝑥𝑣2

Example:

Differentiate f(x) = 𝑥2 + 7 3x − 1

Let u(x) = 𝑥2 + 7
v(x) = 3x−1.
𝑑𝑢𝑑𝑥 = 2x and 𝑑𝑣𝑑𝑥= 3

By using the formula,
𝑑𝑓𝑑𝑥 = v × 𝑑𝑢𝑑𝑥 − u × 𝑑𝑣𝑑𝑥𝑣2
𝑑𝑓𝑑𝑥 = (3x − 1)(2x) − (𝑥2 + 7)(3) (3𝑥−1)2
= 6𝑥2− 2x − 3𝑥2− 21 (3𝑥−1)2
= 3𝑥2 − 2x − 21 (3𝑥−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
𝑑𝑦𝑑𝑥 = 𝑑𝑦𝑑𝑢 × 𝑑𝑢𝑑𝑥

Example:

Differentiate y = (𝑥2− 5)4

Let u = 𝑥2 – 5
y = 𝑢4.
⇒ 𝑑𝑢𝑑𝑥 = 2x and 𝑑𝑦𝑑𝑢 = 4𝑢3.

By using the chain rule formula,

𝑑𝑦𝑑𝑥 = 𝑑𝑦𝑑𝑢 × 𝑑𝑢𝑑𝑥

= 4𝑢3 × 2x
= 4(𝑥2− 5)3 × 2x
8x(𝑥2− 5)3

This is one of the most Important Derivative Rules.

### Rule 8: Exponential Derivative Rules

• If f(x) = 𝑎𝑥 then, f’(x) = ln(a) 𝑎𝑥
• f(x) =𝑒𝑥 then, f’(x) = 𝑒𝑥
• If f(x) = 𝑎𝑔(𝑥) then, f’(x) = ln(a) 𝑎𝑔(𝑥) g’ (x)
• f(x) = 𝑒𝑔(𝑥) then, f’(x) = 𝑒𝑔(𝑥)g’(x)

### Rule 9: Trignometry Derivative rules

• If f(x) = sin(x) then, f’(x) = cos(x)
• When f(x) = cos(x) then, f’(x) = − sin(x)
• If f(x) = tan(x) then, f’(x) = 𝑠𝑒𝑐2(x)
• When f(x) = sec(x) then, f’(x) = sec(x) tan(x)
• If f(x) = cot(x) then, f’(x) = − 𝑐𝑜𝑠𝑒𝑐2(x)
• When f(x) = cosec(x) then, f’(x) = − cosec(x) cot(x)

### Rule 10: Logarithm Derivative rules

• If f(x) = 𝑙𝑜𝑔𝑎(x) then, f’(x) = 1ln(𝑎)𝑥
• f(x) = ln(x) then, f’(x) = 1𝑥
• When f(x) = 𝑙𝑜𝑔𝑎(g(x)) then, f’(x) = 𝑔′(𝑥)ln(𝑎)𝑔(𝑥)
• f(x) = ln(g(x)) then, f’(x) = 𝑔′(𝑥)𝑔(𝑥)