### Question 3. Differentiate each of the following using first principles:

### (i) xsinx

**Solution:**

Given that f(x) = xsinx

By using the formula

We get

=

=

Using the formula

sinc – sind = 2cos((c + d)/2)sin((c – d)/2)

We get

=

As we know that

So,

= 2x × cosx × 1/2 + sinx

= x × cosx + sinx

= sinx + xcosx

### (ii) xcosx

**Solution: **

Given that f(x) = xcosx

By using the formula

We get

=

=

=

=

=

= -xsinx + cosx

### (iii) sin(2x – 3)

**Solution:**

Given that f(x) = sin(2x – 3)

By using the formula

We get

=

=

Using the formula

sinC – sinD = 2cos{C+D}/2sin{C-D}/2

=

As we know that, \lim_{θ\to 0}\frac{sinθ}{θ}=1 so,

= 2cos(2x – 3)

### (iv) √sin2x

**Solution:**

Given that f(x) = √sin2x

By using the formula

We get

=

On multiplying numerator and denominator by

we get

=

=

=

=

=

### (v) sinx/x

**Solution:**

Given that f{x} = sinx/x

By using the formula

We get

=

=

=

=

=

h ⇢ 0 ⇒ h/2 ⇢ 0 and

=

=

### (vi) cosx/x

**Solution:**

Given that f(x) = cosx/x

By using the formula

We get

=

=

=

=

=

=

=

### (vii) x^{2}sinx

**Solution:**

Given that f(x) = x

^{2}sinxBy using the formula

We get

=

=

=

=

= 0 + [2xsinx + x

^{2}cosx]= 2xsinx + x

^{2}cosx

### (viii)

**Solution:**

Given that f(x) =

By using the formula

We get

=

=

=

=

=

### (ix) sinx + cosx

**Solution:**

Given that f(x) = sinx + cosx

By using the formula

We get

=

=

=

=

=

=

= cosx – sinx

### Question 4. Differentiate each of the following using first principles:

### (i) tan^{2}x

**Solution:**

Given that f(x) = tan

^{2}xBy using the formula

We get

=

=

=

=

=

=

=

= 2tanx sec

^{2}x

### (ii) tan(2x + 1)

**Solution:**

Given that f(x) = tan(2x+1)

By using the formula

We get

=

=

=

Multiplying both, numerator and denominator by 2.

=

=

= 2sec

^{2}(2x+1)

### (iii) tan2x

**Solution:**

Given that f(x) = tan2x

By using the formula

We get

=

=

=

=

=

= 2sec

^{2}2x

### (iv) √tanx

**Solution:**

Given that f(x) = √tanx

By using the formula

We get

=

On multiplying numerator and denominator by

We get

=

=

=

=

=

### Question 5. Differentiate each of the following using first principles:

### (i)

**Solution:**

Given that f(x) =

By using the formula

We get

=

=

=

=

=

=

### (ii) cos√x

**Solution:**

Given that f(x) = cos√x

By using the formula

We get

=

=

=

Multiplying numerator and denominator by

=

=

=

=

### (iii) tan√x

**Solution:**

Given that f(x) = tan√x

By using the formula

We get

=

=

=

=

=

=

=

=

### (iv) tanx^{2}

**Solution:**

Given that f(x) = tanx

^{2}By using the formula

We get

=

=

=

=

=

=

=

=

= 2xsec

^{2}x^{2}

### Question 6. Differentiate each of the following using first principles:

### (i) -x

**Solution:**

Given that f(x) = -x

By using the formula

We get

=

=

= -1

### (ii) (-x)^{-1}

**Solution:**

Given that f(x) = (-x)

^{-1}By using the formula

We get

=

=

=

= 1/x

^{2}

### (iii) sin(x + 1)

**Solution:**

Given that f(x) = sin(x+1)

By using the formula

We get

=

=

=

=

=

= cos(x+1)

### (iv) cos(x – π/8)

**Solution:**

We have, f(x)

=cos(x – π/8)By using the formula

We get

=

=

=

=

=

=

= -sin(x + π/8)