#### Page No 301:

#### Question 1:

Evaluate the Given limit:

#### Answer:

#### Question 2:

Evaluate the Given limit:

#### Answer:

#### Question 3:

Evaluate the Given limit:

#### Answer:

#### Question 4:

Evaluate the Given limit:

#### Answer:

#### Question 5:

Evaluate the Given limit:

#### Answer:

#### Question 6:

Evaluate the Given limit:

#### Answer:

Put *x* + 1 = *y* so that *y* → 1 as *x* → 0.

#### Question 7:

Evaluate the Given limit:

#### Answer:

At *x* = 2, the value of the given rational function takes the form.

#### Question 8:

Evaluate the Given limit:

#### Answer:

At *x* = 2, the value of the given rational function takes the form.

#### Question 9:

Evaluate the Given limit:

#### Answer:

#### Question 10:

Evaluate the Given limit:

#### Answer:

At *z* = 1, the value of the given function takes the form.

Put so that *z* →1 as *x* → 1.

#### Question 11:

Evaluate the Given limit:

#### Answer:

#### Question 12:

Evaluate the Given limit:

#### Answer:

At *x* = –2, the value of the given function takes the form.

#### Question 13:

Evaluate the Given limit:

#### Answer:

At *x* = 0, the value of the given function takes the form.

#### Question 14:

Evaluate the Given limit:

#### Answer:

At *x* = 0, the value of the given function takes the form.

#### Page No 302:

#### Question 15:

Evaluate the Given limit:

#### Answer:

It is seen that *x* → π ⇒ (π – *x*) → 0

#### Question 16:

Evaluate the given limit:

#### Answer:

#### Question 17:

Evaluate the Given limit:

#### Answer:

At *x* = 0, the value of the given function takes the form.

Now,

#### Question 18:

Evaluate the Given limit:

#### Answer:

At *x* = 0, the value of the given function takes the form.

Now,

#### Question 19:

Evaluate the Given limit:

#### Answer:

#### Question 20:

Evaluate the Given limit:

#### Answer:

At *x* = 0, the value of the given function takes the form.

Now,

#### Question 21:

Evaluate the Given limit:

#### Answer:

At *x* = 0, the value of the given function takes the form.

Now,

#### Question 22:

#### Answer:

At, the value of the given function takes the form.

Now, put so that.

#### Question 23:

Find *f*(*x*) and*f*(*x*), where *f*(*x*) =

#### Answer:

The given function is

*f*(*x*) =

#### Question 24:

Find *f*(*x*), where *f*(*x*) =

#### Answer:

The given function is

#### Question 25:

Evaluate*f*(*x*), where *f*(*x*) =

#### Answer:

The given function is

*f*(*x*) =

#### Question 26:

Find*f*(*x*), where *f*(*x*) =

#### Answer:

The given function is

#### Question 27:

Find*f*(*x*), where *f*(*x*) =

#### Answer:

The given function is *f*(*x*) =.

#### Question 28:

Suppose *f*(*x*) = and if*f*(*x*) = *f*(1) what are possible values of *a *and *b*?

#### Answer:

The given function is

Thus, the respective possible values of *a* and *b* are 0 and 4.

#### Page No 303:

#### Question 29:

Letbe fixed real numbers and define a function

What is*f*(*x*)? For some compute*f*(*x*).

#### Answer:

The given function is_{.}

#### Question 30:

If *f*(*x*) =.

For what value (s) of a does *f*(*x*) exists?

#### Answer:

The given function is

When *a* < 0,

When *a* > 0

Thus, exists for all *a* ≠ 0.

#### Question 31:

If the function *f*(*x*) satisfies, evaluate.

#### Answer:

#### Question 32:

If. For what integers *m* and *n* does and exist?

#### Answer:

The given function is

Thus, exists if *m* = *n*.

Thus, exists for any integral value of *m* and *n*.

#### Page No 312:

#### Question 1:

Find the derivative of *x*^{2} – 2 at *x* = 10.

#### Answer:

Let *f*(*x*) = *x*^{2} – 2. Accordingly,

Thus, the derivative of *x*^{2} – 2 at *x* = 10 is 20.

#### Question 2:

Find the derivative of 99*x* at *x* = 100.

#### Answer:

Let *f*(*x*) = 99*x*. Accordingly,

Thus, the derivative of 99*x* at *x* = 100 is 99.

#### Question 3:

Find the derivative of *x *at *x *= 1.

#### Answer:

Let* f*(*x*) = *x*. Accordingly,

Thus, the derivative of *x *at *x *= 1 is 1.

#### Question 4:

Find the derivative of the following functions from first principle.

(i) *x*^{3} – 27 (ii) (*x* – 1) (*x *– 2)

(ii) (iv)

#### Answer:

(i) Let *f*(*x*) = *x*^{3} – 27. Accordingly, from the first principle,

(ii) Let *f*(*x*) = (*x* – 1) (*x* – 2). Accordingly, from the first principle,

(iii) Let. Accordingly, from the first principle,

(iv) Let. Accordingly, from the first principle,

#### Question 5:

For the function

Prove that

#### Answer:

The given function is

Thus,

#### Page No 313:

#### Question 6:

Find the derivative offor some fixed real number *a*.

#### Answer:

Let

#### Question 7:

For some constants *a* and *b*, find the derivative of

(i) (*x *– *a*) (*x* –* b*) (ii) (*ax*^{2} + *b*)^{2} (iii)

#### Answer:

(i) Let *f* (*x*) = (*x *– *a*) (*x* – *b*)

(ii) Let

(iii)

By quotient rule,

#### Question 8:

Find the derivative offor some constant *a*.

#### Answer:

By quotient rule,

#### Question 9:

Find the derivative of

(i) (ii) (5*x*^{3} + 3*x *– 1) (*x* – 1)

(iii) *x*^{–3} (5 + 3*x*) (iv) *x*^{5} (3 – 6*x*^{–9})

(v) *x*^{–4} (3 – 4*x*^{–5}) (vi)

#### Answer:

(i) Let

(ii) Let *f* (*x*) = (5*x*^{3} + 3*x *– 1) (*x* – 1)

By Leibnitz product rule,

(iii) Let* f *(*x*) = *x*^{– 3} (5 + 3*x*)

By Leibnitz product rule,

(iv) Let *f* (*x*) = *x*^{5} (3 – 6*x*^{–9})

By Leibnitz product rule,

(v) Let *f *(*x*) = *x*^{–4} (3 – 4*x*^{–5})

By Leibnitz product rule,

(vi) Let *f *(*x*) =

By quotient rule,

#### Question 10:

Find the derivative of cos *x* from first principle.

#### Answer:

Let *f* (*x*) = cos *x*. Accordingly, from the first principle,

#### Question 11:

Find the derivative of the following functions:

(i) sin *x* cos *x* (ii) sec *x* (iii) 5 sec *x* + 4 cos *x*

(iv) cosec *x* (v) 3cot *x* + 5cosec *x*

(vi) 5sin *x* – 6cos *x* + 7 (vii) 2tan *x* – 7sec *x*

#### Answer:

(i) Let* f* (*x*) = sin *x* cos *x*. Accordingly, from the first principle,

(ii) Let* f* (*x*) = sec *x*. Accordingly, from the first principle,

(iii) Let* f* (*x*) = 5 sec *x* + 4 cos *x*. Accordingly, from the first principle,

(iv) Let *f* (*x*) = cosec *x*. Accordingly, from the first principle,

(v) Let *f *(*x*) = 3cot *x* + 5cosec *x*. Accordingly, from the first principle,

From (1), (2), and (3), we obtain

(vi) Let *f* (*x*) = 5sin *x* – 6cos *x* + 7. Accordingly, from the first principle,

(vii) Let *f* (*x*) = 2 tan *x* – 7 sec *x*. Accordingly, from the first principle,

#### Page No 317:

#### Question 1:

Find the derivative of the following functions from first principle:

(i) –*x* (ii) (–*x*)^{–1} (iii) sin (*x* + 1)

(iv)

#### Answer:

(i) Let *f*(*x*) = –*x*. Accordingly,

By first principle,

(ii) Let. Accordingly,

By first principle,

(iii) Let *f*(*x*) = sin (*x* + 1). Accordingly,

By first principle,

(iv) Let. Accordingly,

By first principle,

#### Question 2:

Find the derivative of the following functions (it is to be understood that *a*, *b*, *c*, *d*, *p*, q,* r *and* s* are fixed non-zero constants and *m* and *n* are integers): (*x* + *a*)

#### Answer:

Let *f*(*x*) = *x* + *a*. Accordingly,

By first principle,

#### Question 3:

Find the derivative of the following functions (it is to be understood that *a*, *b*, *c*, *d*, *p*, q,* r *and* s* are fixed non-zero constants and *m* and *n* are integers):

#### Answer:

By Leibnitz product rule,

#### Question 4:

Find the derivative of the following functions (it is to be understood that *a*, *b*, *c*, *d*, *p*, q,* r *and* s* are fixed non-zero constants and *m* and *n* are integers): (*ax* +* b*) (*cx *+ *d*)^{2}

#### Answer:

Let

By Leibnitz product rule,

#### Question 5:

Find the derivative of the following functions (it is to be understood that *a*, *b*, *c*, *d*, *p*, q,* r *and* s* are fixed non-zero constants and *m* and *n* are integers):

#### Answer:

Let

By quotient rule,

#### Question 6:

Find the derivative of the following functions (it is to be understood that *a*, *b*, *c*, *d*, *p*, q,* r *and* s* are fixed non-zero constants and *m* and *n* are integers):

#### Answer:

By quotient rule,

#### Question 7:

*a*, *b*, *c*, *d*, *p*, q,* r *and* s* are fixed non-zero constants and *m* and *n* are integers):

#### Answer:

Let

By quotient rule,

#### Question 8:

*a*, *b*, *c*, *d*, *p*, q,* r *and* s* are fixed non-zero constants and *m* and *n* are integers):

#### Answer:

By quotient rule,

#### Question 9:

*a*, *b*, *c*, *d*, *p*, q,* r *and* s* are fixed non-zero constants and *m* and *n* are integers):

#### Answer:

By quotient rule,

#### Question 10:

*a*, *b*, *c*, *d*, *p*, q,* r *and* s* are fixed non-zero constants and *m* and *n* are integers):

#### Answer:

#### Question 11:

*a*, *b*, *c*, *d*, *p*, q,* r *and* s* are fixed non-zero constants and *m* and *n* are integers):

#### Answer:

#### Question 12:

Find the derivative of the following functions (it is to be understood that *a*, *b*, *c*, *d*, *p*, q,* r *and* s* are fixed non-zero constants and *m* and *n* are integers): (*ax* + *b*)^{n}

#### Answer:

By first principle,

#### Question 13:

Find the derivative of the following functions (it is to be understood that *a*, *b*, *c*, *d*, *p*, q,* r *and* s* are fixed non-zero constants and *m* and *n* are integers): (*ax* + *b*)^{n} (*cx* + *d*)^{m}

#### Answer:

Let

By Leibnitz product rule,

Therefore, from (1), (2), and (3), we obtain

#### Question 14:

Find the derivative of the following functions (it is to be understood that *a*, *b*, *c*, *d*, *p*, q,* r *and* s* are fixed non-zero constants and *m* and *n* are integers): sin (*x* + *a*)

#### Answer:

Let

By first principle,

#### Question 15:

Find the derivative of the following functions (it is to be understood that *a*, *b*, *c*, *d*, *p*, q,* r *and* s* are fixed non-zero constants and *m* and *n* are integers): cosec *x* cot *x*

#### Answer:

Let

By Leibnitz product rule,

By first principle,

Now, let *f*_{2}(*x*) = cosec *x*. Accordingly,

By first principle,

From (1), (2), and (3), we obtain

#### Question 16:

*a*, *b*, *c*, *d*, *p*, q,* r *and* s* are fixed non-zero constants and *m* and *n* are integers):

#### Answer:

Let

By quotient rule,

#### Page No 318:

#### Question 17:

*a*, *b*, *c*, *d*, *p*, q,* r *and* s* are fixed non-zero constants and *m* and *n* are integers):

#### Answer:

Let

By quotient rule,

#### Question 18:

*a*, *b*, *c*, *d*, *p*, q,* r *and* s* are fixed non-zero constants and *m* and *n* are integers):

#### Answer:

Let

By quotient rule,

#### Question 19:

Find the derivative of the following functions (it is to be understood that *a*, *b*, *c*, *d*, *p*, q,* r *and* s* are fixed non-zero constants and *m* and *n* are integers): sin^{n}* x*

#### Answer:

Let *y* = sin^{n}* x*.

Accordingly, for *n* = 1, *y* = sin* x*.

For *n* = 2, *y* = sin^{2}* x*.

For *n* = 3, *y* = sin^{3}* x*.

We assert that

Let our assertion be true for *n* = *k*.

i.e.,

Thus, our assertion is true for *n* = *k* + 1.

Hence, by mathematical induction,

#### Question 20:

*a*, *b*, *c*, *d*, *p*, q,* r *and* s* are fixed non-zero constants and *m* and *n* are integers):

#### Answer:

By quotient rule,

#### Question 21:

*a*, *b*, *c*, *d*, *p*, q,* r *and* s* are fixed non-zero constants and *m* and *n* are integers):

#### Answer:

Let

By quotient rule,

By first principle,

From (i) and (ii), we obtain

#### Question 22:

Find the derivative of the following functions (it is to be understood that *a*, *b*, *c*, *d*, *p*, q,* r *and* s* are fixed non-zero constants and *m* and *n* are integers): *x*^{4} (5 sin *x* – 3 cos *x*)

#### Answer:

Let

By product rule,

#### Question 23:

Find the derivative of the following functions (it is to be understood that *a*, *b*, *c*, *d*, *p*, q,* r *and* s* are fixed non-zero constants and *m* and *n* are integers): (*x*^{2} + 1) cos *x*

#### Answer:

Let

By product rule,

#### Question 24:

Find the derivative of the following functions (it is to be understood that *a*, *b*, *c*, *d*, *p*, q,* r *and* s* are fixed non-zero constants and *m* and *n* are integers): (*ax*^{2} + sin *x*) (*p* + *q* cos *x*)

#### Answer:

Let

By product rule,

#### Question 25:

*a*, *b*, *c*, *d*, *p*, q,* r *and* s* are fixed non-zero constants and *m* and *n* are integers):

#### Answer:

Let

By product rule,

Let. Accordingly,

By first principle,

Therefore, from (i) and (ii), we obtain

#### Question 26:

*a*, *b*, *c*, *d*, *p*, q,* r *and* s* are fixed non-zero constants and *m* and *n* are integers):

#### Answer:

Let

By quotient rule,

#### Question 27:

*a*, *b*, *c*, *d*, *p*, q,* r *and* s* are fixed non-zero constants and *m* and *n* are integers):

#### Answer:

Let

By quotient rule,

#### Question 28:

*a*, *b*, *c*, *d*, *p*, q,* r *and* s* are fixed non-zero constants and *m* and *n* are integers):

#### Answer:

Let

By first principle,

From (i) and (ii), we obtain

#### Question 29:

Find the derivative of the following functions (it is to be understood that *a*, *b*, *c*, *d*, *p*, q,* r *and* s* are fixed non-zero constants and *m* and *n* are integers): (*x* + sec *x*) (*x* – tan *x*)

#### Answer:

Let

By product rule,

From (i), (ii), and (iii), we obtain

#### Question 30:

*a*, *b*, *c*, *d*, *p*, q,* r *and* s* are fixed non-zero constants and *m* and *n* are integers):

#### Answer:

Let

By quotient rule,

It can be easily shown that

Therefore,

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