Question

differentiate 10 sin (4x)​

Answer 1

Answer:

To differentiate means to find the derivative of a function or rate of change of a function with respect to some variable. If a function y is to be differentiated with respect to x, then it will be written as dydx. In the question sin(4x) is a composite function. If f(x) and g(x) are two functions, then f(g(x)) is said to be a composite function. For the given question, f(x)=sinx and g(x)=4x such that f(g(x))=sin(4x). To differentiate sin(4x), we will use the chain rule which is used to differentiate composite functions. The chain rule states that if y=f(g(x)), thendydx=df(g(x))dx=df(g(x))dg(x)×dg(x)dx.

Also, it must be known that the differentiation or derivative of sinx is cosx.

Complete step by step solution:

It is given that y=sin(4x).

Here f(g(x))=sin(4x) where f(x)=sinx and g(x)=4x. On substituting these values in the chain rule of differentiation, we will get

⇒dydx=df(g(x))dx=df(g(x))dg(x)×dg(x)dx

⇒dsin(4x)dx=dsin(4x)d(4x)×d(4x)dx

It must be known that the differentiation or derivative of sinx with respect to x is cosx where x is the argument of the trigonometric function. Therefore the derivative of sin(4x) with respect to 4x is cos(4x). On substituting it, we will get

⇒dsin(4x)dx=cos(4x)×d(4x)dx

Now for differentiating 4x, we will take the constant out, i.e. 4 and use one of the rule of differentiation that states dxndx=nxn−1. Here n=1, so dx1dx=1×x1−1=x0=1. On substituting these values, we will get

⇒dsin(4x)dx=cos(4x)×4×d(x)dx

⇒dsin(4x)dx=cos(4x)×4×1

On further simplifying, we will get

⇒dsin(4x)dx=4cos(4x)

Hence, when we differentiate y=sin(4x) we get 4cos(4x) as the answer.

Note:

If a function y is to be differentiated with respect to x, then it will be written as dydx. But it can also be expressed as Dyas D is sometimes used in place of ddx.

it is usually

Answer 2

What is the differentiation of sin 10?

To apply the Chain Rule, set u u as 10x 10 x . The derivative of sin(u) sin ( u ) with respect to u u is cos(u) cos ( u ) .

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