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Differentiation Formulas Video Lecture | Mathematics (Maths) for JEE Main & Advanced
FAQs on Differentiation Formulas Video Lecture - Mathematics (Maths) for JEE Main & Advanced
| 1. What are the basic differentiation formulas used in JEE? |  |
Ans. The basic differentiation formulas used in JEE include:
- Constant Rule: $\frac{d}{dx}(c) = 0$, where $c$ is a constant.
- Power Rule: $\frac{d}{dx}(x^n) = nx^{n-1}$, where $n$ is a real number.
- Sum/Difference Rule: $\frac{d}{dx}(f(x) \pm g(x)) = \frac{d}{dx}(f(x)) \pm \frac{d}{dx}(g(x))$.
- Product Rule: $\frac{d}{dx}(f(x)g(x)) = f'(x)g(x) + f(x)g'(x)$.
- Quotient Rule: $\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right) = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}$.
| 2. How do we differentiate exponential functions using differentiation formulas in JEE? | |
Ans. To differentiate exponential functions using differentiation formulas in JEE, we use the chain rule. The chain rule states that if we have a function of the form $f(g(x))$, then its derivative is given by $\frac{d}{dx}(f(g(x))) = f'(g(x)) \cdot g'(x)$.
For example, to differentiate the function $f(x) = e^x$, we let $f(x) = e^x$ and $g(x) = x$. Then, using the chain rule, we have $\frac{d}{dx}(e^x) = \frac{d}{dx}(f(g(x))) = f'(g(x)) \cdot g'(x)$. Since the derivative of $e^x$ with respect to $x$ is $1$, we have $\frac{d}{dx}(e^x) = e^x \cdot 1 = e^x$.
| 3. How can we differentiate trigonometric functions using differentiation formulas in JEE? | |
Ans. To differentiate trigonometric functions using differentiation formulas in JEE, we use the derivatives of the basic trigonometric functions. The derivatives of the basic trigonometric functions are as follows:
- $\frac{d}{dx}(\sin x) = \cos x$
- $\frac{d}{dx}(\cos x) = -\sin x$
- $\frac{d}{dx}(\tan x) = \sec^2 x$
- $\frac{d}{dx}(\cot x) = -\csc^2 x$
- $\frac{d}{dx}(\sec x) = \sec x \cdot \tan x$
- $\frac{d}{dx}(\csc x) = -\csc x \cdot \cot x$
For example, to differentiate the function $f(x) = \sin x$, we have $\frac{d}{dx}(\sin x) = \cos x$.
| 4. Can we differentiate logarithmic functions using differentiation formulas in JEE? | |
Ans. Yes, we can differentiate logarithmic functions using differentiation formulas in JEE. The derivative of the natural logarithmic function, $\ln x$, is $\frac{d}{dx}(\ln x) = \frac{1}{x}$.
For example, to differentiate the function $f(x) = \ln x$, we have $\frac{d}{dx}(\ln x) = \frac{1}{x}$.
| 5. Is there a differentiation formula for inverse trigonometric functions in JEE? | |
Ans. Yes, there are differentiation formulas for inverse trigonometric functions in JEE. The derivatives of the inverse trigonometric functions are as follows:
- $\frac{d}{dx}(\arcsin x) = \frac{1}{\sqrt{1-x^2}}$
- $\frac{d}{dx}(\arccos x) = -\frac{1}{\sqrt{1-x^2}}$
- $\frac{d}{dx}(\arctan x) = \frac{1}{1+x^2}$
- $\frac{d}{dx}(\arccot x) = -\frac{1}{1+x^2}$
- $\frac{d}{dx}(\arcsec x) = \frac{1}{|x|\sqrt{x^2-1}}$
- $\frac{d}{dx}(\arccsc x) = -\frac{1}{|x|\sqrt{x^2-1}}$
For example, to differentiate the function $f(x) = \arcsin x$, we have $\frac{d}{dx}(\arcsin x) = \frac{1}{\sqrt{1-x^2}}$.
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