# Derivatives

• Derivative of a function: Let y = f(x) be a function defined on the interval [a, b]. Let for a small increment δx in x, the corresponding increment in the value of ‘y’ be δy. Then \tt \frac{dy}{dx}=\lim_{\delta x \rightarrow 0}\ \frac{\delta y}{\delta x}= \lim_{\delta x \rightarrow 0}\ \frac{f\left(x+\delta x\right)-f\left(x\right)}{\delta x}
• \tt \frac{dy}{dx} represents geometrically the slope of the tangent at the point (x,y) on the curve y = f(x).
• Derivatives of Algebraic functions:
\frac{d}{dx}(x^{n})=nx^{n-1}
\frac{d}{dx}(x)=1
\frac{d}{dx}(x^{2})=2x
\frac{d}{dx}(x^{3})=3x^{2}
\frac{d}{dx}(\sqrt{x})=\frac{1}{2\sqrt{x}}
\frac{d}{dx}\left(\frac{1}{x}\right)=\frac{-1}{x^{2}}
\frac{d}{dx}\left(\frac{1}{x^{2}}\right)=\frac{-2}{x^{3}}
\frac{d}{dx}\left(\frac{1}{x^{3}}\right)=\frac{-3}{x^{4}}
\frac{d}{dx}\left(\frac{1}{\sqrt{x}}\right)=\frac{-1}{2x^{3/2}}
• \tt \frac{d}{dx}\ f\left(x\right)\cdot g\left(x\right)=f\left(x\right)\frac{d}{dx}\ g\left(x\right)+g\left(x\right)\cdot\frac{d}{dx}\ f\left(x\right)
• \tt \frac{d}{dx}\ \frac{f\left(x\right)}{g\left(x\right)}=\frac{g\left(x\right)\cdot\frac{d}{dx}\ f\left(x\right)-f\left(x\right)\cdot\frac{d}{dx}\ g\left(x\right)}{\left[g\left(x\right)\right]^{2}}
• Derivative of Trigonometric functions:
\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)= -cosec^{3} x
\frac{d}{dx}(\sec \ x)=\sec x \ \tan x
\frac{d}{dx}(cosec \ x)=-cosec x \ \cot x

### Part4: View the Topic in this video From 00:12 To 13:52

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1. f'(x)=\frac{d}{dx}f(x)=\lim_{\delta x \rightarrow 0}\frac{f(x+\delta x)-f(x)}{\delta x}

2. \frac{d}{dx}(x^n)=nx^{n-1},x\in R, n \in R

3. (i) \frac{d}{dx}\left\{cf(x)\right\}=c \ \frac{d}{dx} f(x), where c is a constant.

(ii) \frac{d}{dx}\left\{f(x)\pm g(x)\right\}= \frac{d}{dx} f(x)\pm\frac{d}{dx}g(x) (sum and difference rule)

(iii) \frac{d}{dx}\left\{f(x) g(x)\right\}= f(x) \ \frac{d}{dx} \ g(x)+g(x) \ \frac{d}{dx} \ f(x) (product rule)

4. (i) \frac{d}{dx}\left\{\frac{f(x)}{g(x)}\right\}= \frac{g(x) \ \frac{d}{dx} \ f(x)- f(x) \ \frac{d}{dx} \ g(x)}{\left\{g(x)^2\right\}} (quotient rule)

(ii) If \frac{d}{dx}f(x)=\phi(x), then \frac{d}{dx}f(ax+b)=a\phi(ax+b)

(iii) Differentiation of a constant function is zero i.e., \frac{d}{dx}(c)=0.

5. \frac{d}{dx}(\sin x)=\cos x

6. \frac{d}{dx}(\cos x)=-\sin x

7. \frac{d}{dx}(\tan x)=\sec^{2} x,x\neq(2n+1)\frac{\pi}{2}, n\in I

8. \frac{d}{dx}(\cot x)=-cosec^{2} x,x\neq n \pi, n\in I

9. \frac{d}{dx}(\sec x)=\sec x \tan x, x \neq(2n+1)\frac{\pi}{2}, n\in I

10. \frac{d}{dx}(cosec \ x)=- cosec \ x \cot x, x \neq n \pi, n \in I