What is the derivative of #cos(a^3+x^3)#?

3 Answers
Daksh S.
Jun 20, 2016

#-(3x^2)(sin(a^3 + x^3))#

Explanation:

Use chain rule for derivatives.

Consider this as #d/dx (cos(f(x)))# where #f(x)# = #a^3 + x^3#

The answer would be composition of derivatives of #cos(x)# (and putting x as f(x) after differentiating and f(x). Let me demonstrate this in this question.

Derivative of #cos(x)# is #-sin(x)#. Now, let's substitute x with f(x)

So the answer is #(-sin(f(x))xx(d/dx(f(x))))#.

Now, #f(x) = a^3 + x^3#. Assuming a to be a constant, #a^3# is a constant and derivative of a constant is #0#. Derivative of #x^3 # is #3x^2#. I won't explain this because you need to learn this yourself if you can't already figure it out.

So back to the answer.

Ans: #(-sin(a^3 + x^3)xx(d/dx(a^3 + x^3)))#

Final Ans: #-3x^2xxsin(a^3 + x^3)#

Tony B
Jun 20, 2016

Just another way of saying the same thing

#=>(dy)/(dx)=-3x^2sin(a^3+x^3)#

Explanation:

Let #u=a^3+x^3" "->" "(du)/(dx)=3x^2#

Let #y=cos(u)" "->" "(dy)/(du)=-sin(u)#

But #(dy)/(dx)=(du)/(dx)xx(dy)/(du)#

#=>(dy)/(dx)=-3x^2sin(a^3+x^3)#

冠廷 李.
Jun 20, 2016

#d/dx=-sin(a^3+x^3)3x^2#

Explanation:

the main function is #cos(x)#
the sub function is #a^3+x^3#
by the chain rule
main function should differentiate first ，and then differentiate sub function
so
#d/dx=-sin(a^3+x^3)3x^2#

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