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) $f(x) = \cot(x)-\csc(x)$
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: $$f'(x)=-\csc^2(x)-(-\csc(x)\cot(x))=\csc(x)(\cot(x)-\csc(x))$$
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) $f'(x)=csc(x)(cot(x)-csc(x))$