Top Important Derivative Rules to Simplify Your Calculus

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Important Derivative Rules

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.

Derivative(Slope)
Derivative(Slope)

Let’s consider y = f(x) is a function then, the derivative of y can be represented by 𝑑𝑦𝑑π‘₯ or y’ or f’ or 𝑑𝑓𝑑π‘₯.

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 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) = π‘₯𝑛 then f’(x) = nπ‘₯π‘›βˆ’1 β€˜n’ can be a positive, negative, or a fraction. If the function is in the form of π‘₯𝑛, 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‘₯8βˆ’1). Therefore, f’(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 g’(x) = c Β· 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)= 2π‘₯3

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

Step 2: Multiply the constant to the resultant derivative.

Therefore f’(x) = 6π‘₯2

Rule 4: Sum or Difference Rule.

If f(x) = h(x) Β± g(x) then, 𝑑𝑓𝑑π‘₯ = π‘‘β„Žπ‘‘π‘₯ Β± 𝑑𝑔𝑑π‘₯

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

Example:

Differentiate f(x) = 3π‘₯2 βˆ’ 7x.

Here, let us consider k(x) = 3π‘₯2 ,
g(x) = 7x
=> π‘‘π‘˜π‘‘π‘₯ = 6x and 𝑑𝑔𝑑π‘₯ = 7.

Therefore, f’(x) = 6x βˆ’ 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)= 𝑑𝑒𝑑π‘₯ , v’(x)= 𝑑𝑣𝑑π‘₯

Example:

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

Let u(x) = 6π‘₯2 + 2x
v(x) = π‘₯3 + 1.
οƒ° 𝑑𝑒𝑑π‘₯ = 12x + 2 and 𝑑𝑣𝑑π‘₯ = 3π‘₯2 .

By using formula,
𝑑𝑓𝑑π‘₯= u Γ— 𝑑𝑣𝑑π‘₯ + v Γ— 𝑑𝑒𝑑π‘₯

Therefore, 𝑑𝑓𝑑π‘₯ = (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) =𝑒(π‘₯)𝑣(π‘₯) , where u(x) and v(x) are both functions of x then,
𝑑𝑓𝑑π‘₯ = v Γ— 𝑑𝑒𝑑π‘₯ βˆ’ u Γ— 𝑑𝑣𝑑π‘₯𝑣2

Example:

Differentiate f(x) = π‘₯2 + 7 3x βˆ’ 1

Let u(x) = π‘₯2 + 7
v(x) = 3xβˆ’1.
𝑑𝑒𝑑π‘₯ = 2x and 𝑑𝑣𝑑π‘₯= 3

By using the formula,
𝑑𝑓𝑑π‘₯ = v Γ— 𝑑𝑒𝑑π‘₯ βˆ’ u Γ— 𝑑𝑣𝑑π‘₯𝑣2
𝑑𝑓𝑑π‘₯ = (3x βˆ’ 1)(2x) βˆ’ (π‘₯2 + 7)(3) (3π‘₯βˆ’1)2
= 6π‘₯2βˆ’ 2x βˆ’ 3π‘₯2βˆ’ 21 (3π‘₯βˆ’1)2
= 3π‘₯2 βˆ’ 2x βˆ’ 21 (3π‘₯βˆ’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
𝑑𝑦𝑑π‘₯ = 𝑑𝑦𝑑𝑒 Γ— 𝑑𝑒𝑑π‘₯

Example:

Differentiate y = (π‘₯2βˆ’ 5)4

Let u = π‘₯2 – 5
y = 𝑒4.
β‡’ 𝑑𝑒𝑑π‘₯ = 2x and 𝑑𝑦𝑑𝑒 = 4𝑒3.

By using the chain rule formula,

𝑑𝑦𝑑π‘₯ = 𝑑𝑦𝑑𝑒 Γ— 𝑑𝑒𝑑π‘₯

= 4𝑒3 Γ— 2x
= 4(π‘₯2βˆ’ 5)3 Γ— 2x
8x(π‘₯2βˆ’ 5)3

This is one of the most Important Derivative Rules.

Rule 8: Exponential Derivative Rules

  • If f(x) = π‘Žπ‘₯ then, f’(x) = ln(a) π‘Žπ‘₯
  • f(x) =𝑒π‘₯ then, f’(x) = 𝑒π‘₯
  • If f(x) = π‘Žπ‘”(π‘₯) then, f’(x) = ln(a) π‘Žπ‘”(π‘₯) g’ (x)
  • f(x) = 𝑒𝑔(π‘₯) then, f’(x) = 𝑒𝑔(π‘₯)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) = βˆ’ 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) = βˆ’ π‘π‘œπ‘ π‘’π‘2(x)
  • When f(x) = cosec(x) then, f’(x) = βˆ’ cosec(x) cot(x)

Rule 10: Logarithm Derivative rules

  • If f(x) = π‘™π‘œπ‘”π‘Ž(x) then, f’(x) = 1ln(π‘Ž)π‘₯
  • f(x) = ln(x) then, f’(x) = 1π‘₯
  • When f(x) = π‘™π‘œπ‘”π‘Ž(g(x)) then, f’(x) = 𝑔′(π‘₯)ln(π‘Ž)𝑔(π‘₯)
  • f(x) = ln(g(x)) then, f’(x) = 𝑔′(π‘₯)𝑔(π‘₯)

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

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