Calculus is the study of derivatives. It is based on the rate of change for example if we consider velocity = $\frac{dx}{dt}$. Where, x is the distance, which keeps changing with time.

In the above example we see that velocity is different at different time intervals. Hence, we infer this as rate of change of time with respect to distance.

So we always try to say that "A variable is being differentiated with respect to some other variable"

We shall try to put some formulas that is being used regularly.

1. $\frac{d}{dx}$ ( x^{n}) = nx^{n-1}

2. $\frac{d}{dx}$ (sinx) =cosx

3. $\frac{d}{dx}$(cosx)=-sinx

4. $\frac{d}{dx}$(tanx) =sec^{2} x

5. $\frac{d}{dx}$(cotx) =-cosec^{2}x

6. $\frac{d}{dx}$(secx) = secx tanx

7. $\frac{d}{dx}$(cosecx)= -cosecx cotx

8.

9.

10.

**Find the derivative of x ^{2} + 3 sinx - tanx.**

Answer: $\frac{d}{dx}$ ( x

We know that the derivative of x

so 2x

$\frac{d}{dx}$(sinx) = cosx

$\frac{d}{dx}$( tanx) = sec

Answer: 2x + 3cosx -sec

**Find the derivative of e ^{3}x - e ^{2}x + cotx**

Answer: $\frac{d}{dx}$ (e

: 3 e

** The distance travelled by the car is given by the equation X = 3t ^{3}-t^{2} find the velocity of the car at the instant t= 1sec**

We all know that velocity = $\frac{dx}{dt}$

= $\frac{d}{dt}$ (3t

= 3(3t

= 9 t

= 9[1]

= 9-2

= 7m/sec.

** The distance travelled by the car is given by the equation X = 3t ^{4}-t^{2} find the velocity of the car at the instant t= 1sec**

We all know that velocity = $\frac{dx}{dt}$

= $\frac{d}{dt}$ (3t

= 4(3t

= 12 t

= 12[1]

= 12-2

= 10m/sec.

**The distance travelled by the car is given by the equation X = t ^{3}-t^{4} find the velocity of the car at the instant t= 2sec**

We all know that velocity = $\frac{dx}{dt}$

= $\frac{d}{dt}$ (t

= 3(t

= 3 t

= 3[2]

= 12-24

= -12 m/sec. [the vehicle is coming to halt]

**The distance travelled by the car is given by the equation X = 3t ^{4}-t^{2} find the velocity of the car at the instant t= 2sec**

We all know that velocity = $\frac{dx}{dt}$

= $\frac{d}{dt}$ (3t

= 4(3t

= 12 t

= 12[2]

= 48-4

= 44 m/sec.

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