Question

12. Find the derivative of (from first principle)
sin 4x

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Answer 1

Answer:

Given: f(x)=sin4x

To find : derivative of f(x)=sin4x from 1st principle

Step-by-step explanation:

f(x)=sin4x

f'(x) = 4Cos4x

using 1st principle

f'(x) = Lim h→ 0 ( f(x + h) - f(x) )/h

= Lim h→ 0 (sin(4x + 4h) - Sin(4x)) /h

Using SinA - SinB = 2Cos((A + B)/2) Sin((A - B)/2)

= Lim h→ 0 2Cos(4x + 2h)Sin(2h) / h

= Lim h→ 0 2Cos(4x + 2h) 2 Sin(2h) / 2h

= Lim h→ 0 4Cos(4x + 2h) Sin(2h) / 2h

using Lim h→ 0 Sin(2h) / 2h = 1

= Lim h→ 0 4Cos(4x + 2h)

= 4Cos4x

f'(x) = 4Cos4x

Answer 2

Step 1:

Identify the inner and outer functions.

Here, the outer function is ‘sin’ and inner function is 4x.

Step 2:

Differentiate the outer function first. And write the inner function as it is.

12

The outer function is Sin.

Therefore,

Differtial of sin is cos. And on writing the inner function as it is,

we get,

Cos4x

Step 3:

Differentiate the inner function, and then multiply it to the outer function’s derivative.

Differentiating inner function 4x,

Derivative of 4x is 4.

Now, Multiplying inner function's derivation with outer function's derivative.

i.e., 4cos4x.

Hence,

The derivative of sin4x is 4cos4x.....