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What is the derivative of sinx(sinx+cosx)?

Answer
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Hint: We use Product – Rule to find the derivative of function sinx(sinx+cosx).
The product rule helps us to differentiate between two or more of the functions in a given function.
If u and v are the two given function of x then the product rule is given by the following formula:
d(uv)dx=udvdx+vdudx. After product rule we use different trigonometric identities such as cos2x=cos2xsin2x and sin2x=2sinxcosx to solve the given problem.

Complete step by step answer:
The given function is the product of two functions sinx and (sinx+cosx).
We use product rule to find the derivative, that means first we multiply the first function by the derivative of the second function and the second function is multiplied by the derivative of the first function and add them.
So, we can write it as
sinx(d(sinx+cosx)dx)+(sinx+cosx)(d(sinx)dx)(1)
We know that the derivative of sinx=cosx and the derivative of cosx=sinx.
So, d(sinx+cosx)dx=(cosxsinx) and dsinxdx=cosx
Put these values in equation 1. We get,
sinx(cosxsinx)+(sinx+cosx)(cosx)
Simplifying the above equation. We get,
sinxcosxsin2x+sinxcosx+cos2x
2sinxcosx+cos2xsin2x
We know that cos2x=cos2xsin2x and sin2x=2sinxcosx.
Write these values in the above equation. We get,
sin2x+cos2x
Hence, the derivative of the function sinx(sinx+cosx) is sin2x+cos2x.

Note:
To find the derivative of the given function we can also use chain rule. First we simplify the expression by opening brackets and we will get a composite function. Chain rule is used where the function is composite, we can denote chain rule by f.g, where f and g are two functions. Chain rule states that the derivative of a composite function can be taken as the derivative of the outer function which we multiply by the derivative of the inner function.