Differentiation Formulas

A Differentiation formulas list has been provided here for students so that they can refer to these to solve problems based on differential equations. This is one of the most important topics in higher-class Mathematics. The general representation of the derivative is d/dx.

This formula list includes derivatives for constant, trigonometric functions, polynomials, hyperbolic, logarithmic functions, exponential, inverse trigonometric functions etc. Based on these, there are a number of examples and problems present in the syllabus of Class 11 and 12, for which students can easily write answers.

Differentiation Formulas List

In all the formulas below, f’ and g’ represents the following:

• \(\begin{array}{l} \frac{d(f(x))}{dx} = f'(x)\end{array} \)
• \(\begin{array}{l}\frac{d(g(x))}{dx}= g'(x) \end{array} \)

Both f and g are the functions of x and are differentiated with respect to x. We can also represent dy/dx = D_x y. Some of the general differentiation formulas are;

1. Power Rule: (d/dx) (xⁿ ) = nx^n-1
2. Derivative of a constant, a:  (d/dx) (a) = 0
3. Derivative of a constant multiplied with function f: (d/dx) (a. f) = af’
4.  Sum Rule: (d/dx) (f ± g) = f’ ± g’
5. Product Rule: (d/dx) (fg)= fg’ + gf’ 
6. Quotient Rule:
   \(\begin{array}{l}\frac{d}{dx}(\frac{f}{g})= \frac{gf’ – fg’}{g^2}\end{array} \)

Differentiation Formulas for Trigonometric Functions

Trigonometry is the concept of the relationship between angles and sides of triangles. Here, we have 6 main ratios, such as, sine, cosine, tangent, cotangent, secant and cosecant. You must have learned about basic trigonometric formulas based on these ratios. Now let us see the formulas for derivatives of trigonometric functions and hyperbolic functions.

1. \(\begin{array}{l}\frac{d}{dx} (sin~ x)= cos\ x\end{array} \)
2. \(\begin{array}{l}\frac{d}{dx} (cos~ x)= – sin\ x\end{array} \)
3. \(\begin{array}{l}\frac{d}{dx} (tan ~x)= sec^{2} x\end{array} \)
4. \(\begin{array}{l}\frac{d}{dx} (cot~ x = -cosec^{2} x\end{array} \)
5. \(\begin{array}{l}\frac{d}{dx} (sec~ x) = sec\ x\ tan\ x\end{array} \)
6. \(\begin{array}{l}\frac{d}{dx} (cosec ~x)= -cosec\ x\ cot\ x\end{array} \)
7. \(\begin{array}{l}\frac{d}{dx} (sinh~ x)= cosh\ x\end{array} \)
8. \(\begin{array}{l}\frac{d}{dx} (cosh~ x) = sinh\ x\end{array} \)
9. \(\begin{array}{l}\frac{d}{dx} (tanh ~x)= sech^{2} x\end{array} \)
10. \(\begin{array}{l}\frac{d}{dx} (coth~ x)=-cosech^{2} x\end{array} \)
11. \(\begin{array}{l}\frac{d}{dx} (sech~ x)= -sech\ x\  tanh\ x\end{array} \)
12. \(\begin{array}{l}\frac{d}{dx} (cosech~ x ) = -cosech\ x\ coth\ x\end{array} \)

Differentiation Formulas for Inverse Trigonometric Functions

Inverse trigonometry functions are the inverse of trigonometric ratios. Let us see the formulas for derivatives of inverse trigonometric functions.

1. \(\begin{array}{l}\frac{d}{dx}(sin^{-1}~ x)=\frac{1}{\sqrt{1 – x^2}}\end{array} \)
2. \(\begin{array}{l}\frac{d}{dx}(cos^{-1}~ x) = -\frac{1}{\sqrt{1 – x^2}}\end{array} \)
3. \(\begin{array}{l}\frac{d}{dx}(tan^{-1}~ x) = \frac{1}{1 + x^2}\end{array} \)
4. \(\begin{array}{l}\frac{d}{dx}(cot^{-1}~ x) = -\frac{1}{1 + x^2}\end{array} \)
5. \(\begin{array}{l}\frac{d}{dx}(sec^{-1} ~x) = \frac{1}{|x|\sqrt{x^2 – 1}}\end{array} \)
6. \(\begin{array}{l}\frac{d}{dx}(cosec^{-1}~x) = -\frac{1}{|x|\sqrt{x^2 – 1}}\end{array} \)

Other Differentiation Formulas

1. \(\begin{array}{l}\frac{d}{dx}(a^{x}) = a^{x} ln a\end{array} \)
2. \(\begin{array}{l}\frac{d}{dx}(e^{x}) = e^{x}\end{array} \)
3. \(\begin{array}{l}\frac{d}{dx}(log_a~ x) = \frac{1}{(ln~ a)x}\end{array} \)
4. \(\begin{array}{l}\frac{d}{dx}(ln~ x) = 1/x\end{array} \)
5. Chain Rule:
   \(\begin{array}{l}\frac{dy}{dx}= \frac{dy}{du}\times \frac{du}{dx}= \frac{dy}{dv}\times \frac{dv}{du}\times \frac{du}{dx}\end{array} \)

Differentiation Formulas PDF

In this section, we have provided a PDF on differentiation formulas for easy access. This PDF includes the derivatives of some basic functions, logarithmic and exponential functions. Apart from these formulas, PDF also covered the derivatives of trigonometric functions and inverse trigonometric functions as well as rules of differentiation. All these formulas help in solving different questions in calculus quickly and efficiently.

Download Differentiation Formulas PDF Here

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