# Differentiation Formulas

Differentiation forms the basis of Calculus. Solving Complex Functions using Derivative Definition seems to be an almost impossible task. Luckily, you need not bother as we have compiled the large collection of Derivative Formulas. Have a glance at the Differentiation Formulas List which will be of great help to solve your problems of complex functions and limits.

## List of Differentiation Formulae

We have listed the Differentiation Formulas List so that students can make use of them while solving Problems on Differential Equations. Formula Sheet of Derivates includes numerous formulas covering derivative for constant, trigonometric functions, hyperbolic, exponential, logarithmic functions, polynomials, inverse trigonometric functions, etc. Apply the Differentiation Formulae provided in your problems and get the results easily.

1. Differential Coefficient

The derivative or differential coefficient of y with respect to x and it is dy

denoted by \(\frac{d y}{d x}\), y’, y_{1} or Dy

so, \(\frac{d y}{d x}=\lim _{8 x \rightarrow 0} \frac{\delta y}{\delta x}\)

2. Differentiability of a function

A function f(x) is said to be differentiable at a point of its domain if it has a finite derivative at that point. Thus f(x) is differentiable at x = a

\(\frac{d y}{d x}=\lim _{h \rightarrow 0} \frac{f(a-h)-f(a)}{-h}=\lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h}\)

⇒ f'(a – 0) = f'(a + 0)

⇒ left-hand derivative = right-hand derivative.

Thus function f is said to be differentiable if left hand derivative & right hand derivative both exist finitely and are equal.

If f(x) is differentiable then its graph must be smooth i.e. there should be no break or corner.

Note:

(i) Every differentiable function is necessarily continuous but every continuous function is not necessarily differentiable i.e. Differentiability ⇒ continuity but continuity ⇏ differentiability

(ii) For any curve y = f(x), if at any point \(\frac{d y}{d x}\) = 0 or does not exist then, the point is called “critical point”.

3. Differentiability in an interval

(a) A function ffx) is said to be differentiable in an open interval (a, b), if it is differentiable at every point of the interval.

(b) A function f(x) is differentiable in a closed interval [a, b] if it is

- Differentiable at every point of interval (a, b)
- Right derivative exists at x = a
- Left derivative exists at x = b.

4. Differential coefficient of some standard function

- \(\frac{d}{d x}\)(constant) = 0
- \(\frac{d}{d x}\)(ax) = a
- \(\frac{d}{d x}\)(x
^{n}) = nx^{n-1} - \(\frac{d}{d x}\) e
^{x}= e^{x} - \(\frac{d}{d x}\)(a
^{x}) = a^{x}log_{e}a - \(\frac{d}{d x}\)(log
_{e}x) = \(\frac{1}{x}\) - \(\frac{d}{d x}\)(log
_{a}x) = \(\frac{1}{xloga}\) - \(\frac{d}{d x}\)(sin x) = cos x
- \(\frac{d}{d x}\)(cos x) = -sin x
- \(\frac{d}{d x}\)(tan x) = sec
^{2}x - \(\frac{d}{d x}\)(cot x) = -cosec
^{2}x - \(\frac{d}{d x}\)(sec x) = sec x tan x
- \(\frac{d}{d x}\)(cosec x) = -cosec x cot x
- \(\frac{d}{d x}\)(sin
^{-1}x) = \(\frac{1}{\sqrt{1-x^{2}}}\), -1 < x < 1 - \(\frac{d}{d x}\)(cos
^{-1}x) = –\(\frac{1}{\sqrt{1-x^{2}}}\), -1 < x < 1 - \(\frac{d}{d x}\)(tan
^{-1}x) = \(\frac{1}{1+x^{2}}\) - \(\frac{d}{d x}\)(cot
^{-1}x) = –\(\frac{1}{1+x^{2}}\) - \(\frac{d}{d x}\)(sec
^{-1}x) = \(\frac{1}{x \sqrt{x^{2}-1}}\), |x| > 1 - \(\frac{d}{d x}\)(cosec
^{-1}x) = –\(\frac{1}{x \sqrt{x^{2}-1}}\) - \(\frac{d}{d x}\)(sinh x) = cosh x
- \(\frac{d}{d x}\)(cosh x) = sinh x
- \(\frac{d}{d x}\)(tanh x) = sech
^{2}x - \(\frac{d}{d x}\)(coth x) = -cosech
^{2}x - \(\frac{d}{d x}\)(sech x) = -sech x tanh x
- \(\frac{d}{d x}\)(cosech x) = -cosech x coth x
- \(\frac{d}{d x}\)(sinh
^{-1}x) = \(\frac{1}{\sqrt{1+x^{2}}}\) - \(\frac{d}{d x}\)(cosh
^{-1}x) = \(\frac{1}{\sqrt{x^{2}-1}}\), x > 1 - \(\frac{d}{d x}\)(tanh
^{-1}x) = \(\frac{1}{1-x^{2}}\) - \(\frac{d}{d x}\)(coth
^{-1}x) = \(\frac{1}{x^{2}-1}\), |x| > 1 - \(\frac{d}{d x}\)(sech
^{-1}x) = –\(\frac{1}{x \sqrt{1-x^{2}}}\), (0 < x < 1) - \(\frac{d}{d x}\)(cosech
^{-1}x) = –\(\frac{1}{|x| \sqrt{x^{2}+1}}\), x ≠ 0 - \(\frac{d}{d x}\) (e
^{ax}sin bx) = e^{ax}(a sin bx + b cos bx)

= \(\sqrt{a^{2}+b^{2}}\) e^{ax}sin (bx + tan^{-1}b/a) - \(\frac{d}{d x}\) (e
^{ax}cos bx) = e^{ax}(a cos bx – b sin bx)

= \(\sqrt{a^{2}+b^{2}}\) e^{ax}cos (bx + tan^{-1}b/a)

5. Some theorems on Differentiation

Theorem I \(\frac{d}{d x}\) [kf(x)] = k \(\frac{d}{d x}\) [f(x)], where k is a constant

Theorem II \(\frac{d}{d x}\) [f_{1}(x) ± f_{2}(x) ± f_{3}(x) ±….]

= \(\frac{d}{d x}\)[f_{1}(x)] ± \(\frac{d}{d x}\) [f_{2}(x)] ± \(\frac{d}{d x}\) [f_{3}(x)] + …….

Theorem III

\(\frac{d}{d x}\)[f(x).g(x)] = f(x)\(\frac{d}{d x}\)[g(x)] + g(x)\(\frac{d}{d x}\)[f(x)]

Theorem IV

\(\frac{\mathrm{d}}{\mathrm{dx}}\left[\frac{\mathrm{f}(\mathrm{x})}{\mathrm{g}(\mathrm{x})}\right]=\frac{\mathrm{g}(\mathrm{x}) \frac{\mathrm{d}}{\mathrm{dx}}[\mathrm{f}(\mathrm{x})]-\mathrm{f}(\mathrm{x}) \frac{\mathrm{d}}{\mathrm{dx}}[\mathrm{g}(\mathrm{x})]}{[\mathrm{g}(\mathrm{x})]^{2}}\)

Theorem V

(CHAIN RULE)

Derivative of the function of the function. If ‘y’ is a function of ‘t’ and ‘t’ is function of ‘x’ then

\(\frac{d y}{d x}=\frac{d y}{d t} \cdot \frac{d t}{d x}\)

Theorem VI

Derivative of parametric equations.

If x = Φ(t), y = φ(t) then

\(\frac{d y}{d x}=\frac{d y / d t}{d x / d t}=\frac{\psi^{\prime}(t}{d^{\prime}(t)}\)

Theorem VII

Derivative of a function with respect to another fimction.If f(x) and g(x) are two functions of a variables x, then differentiation of f(x) w.r.t. g(x) can be written as

\(\frac{d[f(x)]}{d[g(x)]}=\frac{d}{d x}[f(x)] / \frac{d}{d x}[g(x)]\)

6. Differentiation of Implicit Functions

If in an equation, x and y both occurs together i.e. f(x, y) = 0 and thisequation can not be solved either for y or x, then y (or x) is called the implicit function of x (or y).

For example x^{3} + y^{3} + 3axy + c = 0, x^{y} + y^{x} = a^{b} etc.

Working rule for finding the derivative

First Method:

(i) Differentiate every term of f(x,y) = 0 with respect to x.

(ii) Collect the coefficients of \(\frac{d y}{d x}\) and obtain the value of \(\frac{d y}{d x}\)

Second Method:

If f(x, y) = constant, then \(\frac{d y}{d x}=-\frac{\partial f / \partial x}{\partial f / \partial y}\)

where \(\frac{\partial f}{\partial x}\) and \(\frac{\partial f}{\partial y}\) are partial differential coefficients of f(x, y) with respect to x and y respectively.

7. Differentiation of f(x), g(x) type functions

When base and power both are the functions of x i.e. the function is of the form [f(x)]^{g(x)}.

y = [f(x)]^{g(x)}

log y = g(x) log [f(x)]

\(\frac{1}{y} \cdot \frac{d y}{d x}=\frac{d}{d x}\)g(x).log[f(x)]

\(\frac{d y}{d x}=[f(x)]^{g(x)} \cdot\left\{\frac{d}{d x}[g(x) \log f(x)]\right\}\)

i.e. \(\frac{d y}{d x}=Q \cdot \frac{d}{d x}\)(power x log base)

8. Differentiation by trigonometrical substitutions

Some times before differentiation, we reduce the given function in a simple form using suitable trigonometrical or algebric transformations.

Function | Substitution |

(i) \( \sqrt{a^{2}-x^{2}}\) | x = a sin θ or a cos θ |

(ii) \( \sqrt{x^{2}+a^{2}}\) | x = a tan θ or a cot θ |

(iii) \( \sqrt{x^{2}-a^{2}}\) | x = a sec θ or a cosec θ |

(iv) \( \sqrt{\frac{a-x}{a+x}}\) | x = a cos 2θ |

(v) \( \sqrt{\frac{a^{2}-x^{2}}{a^{2}+x^{2}}}\) | x^{2} = a^{2} cos 2θ |

(vi) \( \sqrt{a x-x^{2}}\) | x = a sin^{2 }θ |

(vii) \( \sqrt{\frac{x}{a+x}}\) | x = a tan^{2 }θ |

(viii) \( \sqrt{\frac{x}{a-x}}\) | x = a sin^{2 }θ |

(ix) \( \sqrt{(x-a)(x-b)}\) | x = a sec^{2 }θ – b tan^{2 }θ |

(x) \( \sqrt{(x-a)(b-x)}\) | x = a cos^{2 }θ + b sin^{2 }θ |

9. Differentiation of Infinite Series

(i) If y = \(\sqrt{f(x)+\sqrt{f(x)+\sqrt{f(x)+\ldots . \infty}}}\) then

⇒ y = \(\sqrt{f(x)+y}\)

⇒ y^{2} = f(x) + y

⇒ 2y\(\frac{d y}{d x}=f^{\prime}(x)+\frac{d y}{d x}\)

∴ \(\frac{d y}{d x}=\frac{f^{\prime}(x)}{2 y-1}\)

(ii) If y = f(x)^{f(x)f(x)……∞} then y = f(x)^{y}

∴ log y = y log [f(x)]

\(\frac{1}{y} \frac{d y}{d x}=\frac{y \cdot f^{\prime}(x)}{f(x)}+\log f(x) \cdot\left(\frac{d y}{d x}\right)\)

∴ \(\frac{d y}{d x}=\frac{y^{2} f^{\prime}(x)}{f(x)[1-y \log f(x)]}\)

(iii) If y = f(x) + \(\frac{1}{f(x)}\)_{+\(\frac{1}{f(x)}\)+\(\frac{1}{f(x)}\)……∞}

then \(\frac{d y}{d x}=\frac{y f^{\prime}(x)}{2 y-f(x)}\).

10. n^{th} Derivatives of some standard Functions

(i) \(\frac{d^{\mathrm{n}}}{d x^{n}}\) sin(ax + b) = a^{n} sin[\(\frac{n \pi}{2}\) + ax + b]

(ii) \(\frac{d^{\mathrm{n}}}{d x^{n}}\) cos(ax + b) = a^{n} cos [\(\frac{n \pi}{2}\) + ax + b]

(iii) \(\frac{d^{\mathrm{n}}}{d x^{n}}\) (ax + b)^{m} = \(\frac{m !}{(m-n) !}\) a^{n} (ax + b)^{m-n}, where m > n

(iv) \(\frac{d^{\mathrm{n}}}{d x^{n}}\) (log(ax + b)) = \(\frac{(-1)^{n-1}(n-1) ! a^{n}}{(a x+b)^{n}}\)

(v) \(\frac{d^{\mathrm{n}}}{d x^{n}}\) (e^{ax}) = a^{n} e^{ax}

(vi) \(\frac{d^{n}\left(a^{x}\right)}{d x^{n}}\) = a^{x} (log a)^{n}

(vii) \(\frac{d^{\mathrm{n}}}{d x^{n}}\)(e^{ax} sin(bx + c)) = r^{n}e^{ax} sin(bx + c + nΦ)

where r = \(\sqrt{a^{2}+b^{2}}\); Φ = tan^{-1}\(\frac{b}{a}\)

(viii) \(\frac{d^{\mathrm{n}}}{d x^{n}}\)(e^{ax} cos(bx + c)) = r^{n}e^{ax} cos(bx + c + nΦ)

where r = \(\sqrt{a^{2}+b^{2}}\); Φ = tan^{-1}\(\frac{b}{a}\)