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) $$f(x) = \cot(x)-\csc(x)$$
See development and solution
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) $$f'(x)=csc(x)(cot(x)-csc(x))$$