Question

What is the derivative of $\sin (-2x)$ ?

Answer 1

First, we need to analyze the given information which is in the trigonometric form.
$\bullet$ We can equate the given expression into some form and then we can differentiate using the derivatives of basic functions and applying the chain rule of differentiation.
$\bullet$ The trigonometric functions are useful whenever trigonometric functions are involved in an expression or an equation and these identities are useful whenever expressions involving trigonometric functions need to be simplified.
Formula used:
Chain rule of differentiation $\dfrac{d}{{dx}}(f(g(x)) = {f^1}(g(x)) \times {g^1}(x)$

Complete step-by-step solution:
Since from the given that we have $\sin (-2x)$ and we need to find its derivative part. Also, note that $2$ is a constant function and $x$ is the variable to the given sine function.
Thus, using the chain rule of the differentiation, we have
$\dfrac{d}{{dx}}(f(g(x)) = {f^1}(g(x)) \times {g^1}(x)$
$\Rightarrow \dfrac{d}{{dx}}(\sin (-2x)) = \dfrac{{d(\sin (-2x))}}{{d(-2x)}} \times \dfrac{{d(-2x)}}{{dx}}$
Since we know the derivative of the $\sin x$ with respect to the variable $x$ is $\dfrac{d}{{dx}}(\sin x) = \cos x$
Now we proceed to further steps using the formula given
$\dfrac{d}{{dx}}(\sin (-2x)) = \dfrac{{d(\sin (-2x))}}{{d(-2x)}} \times \dfrac{{d(-2x)}}{{dx}}$ $\Rightarrow \dfrac{{d(\sin (-2x))}}{{d(-2x)}} \times -2\dfrac{{dx}}{{dx}}$
Since the derivative of the constant function will not change.
Then we get,
$\dfrac{d}{{dx}}(\sin (-2x)) = \dfrac{{d(\sin (-2x))}}{{d(-2x)}} \times \dfrac{{d(-2x)}}{{dx}}$ $\Rightarrow \cos (-2x) \times -2$
Hence, we have $\dfrac{d}{{dx}}(\sin (-2x)) = -2\cos (-2x)= -2\cos(2x)$ As we know $\cos(-\theta)=\cos(\theta)$
Thus, the derivative of the given trigonometric function $\sin (-2x)$ is $-2\cos (2x)$.

Note: The concept used in the given problem is the chain rule is the main thing. We must know the derivatives of the basic functions like a sine.
Differentiation is defined as the derivative of independent variable value and can be used to calculate features in an independence variance per unit modification.
In differentiation, the derivative of $x$ raised to the power is denoted by $\dfrac{d}{{dx}}\left( {{x^n}} \right) = n{x^{n - 1}}$ .
Differentiation and integration are inverse processes like a derivative of $\dfrac{{d({x^2})}}{{dx}} = 2x$ and the integration is $\int {2xdx = \dfrac{{2{x^2}}}{2}} \Rightarrow {x^2}$
In total there are six trigonometric values which are sine, cos, tan, sec, cosec, cot while all the values have been relation like $\dfrac{{\sin }}{{\cos }} = \tan$and $\tan = \dfrac{1}{{\cot }}$.