Problems from Derivative of trigonometric functions

Find the derivative of the following functions:

a) $$f(x)=x+\sin(x)$$

b) $$f(x) = 5\cos(x) +16 x^2$$

c) $$f(x)=\arctan(x)+ \cos(x)-x^6$$

d) $$f(x) = \cot(x)-\csc(x)$$

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Development:

a) Using the rule of the sum, we recognize $$g(x)=x$$ and $$h(x) =\sin(x)$$, and therefore, $$$f'(x)=1+\cos(x)$$$

b) In this case, $$g(x)=5\cos(x)$$ and $$h(x)=16$$. Therefore, $$$f'(x)=5(-\sin(x))+16(2x)=32x-5\sin(x)$$$

c) We recognize now three different functions; we apply the rule of the sum and obtain: $$$f'(x)=\dfrac{1}{1+x^2}-\sin(x)-6x^5$$$

d) By applying the rule of the sum: $$$f'(x)=-\csc^2(x)-(-\csc(x)\cot(x))=\csc(x)(\cot(x)-\csc(x))$$$

Solution:

a) $$f'(x)=1+\cos(x)$$

b) $$f'(x)=32x-5\sin(x)$$

c) $$f'(x)=\dfrac{1}{1+x^2}-\sin(x)-6x^5$$

d) $$f'(x)=csc(x)(cot(x)-csc(x))$$

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