Question

What is the derivative of\[\cos \,{x^3}\]?

Answer 1

Hint: The derivative is a tool used in mathematics (particularly in differential calculus) to depict instantaneous rate of change, or the amount by which a function changes at a given point. It is the slope of the tangent line at a point on a graph for functions that work on real numbers.

Complete step-by-step solution:
To find the derivative of \[\cos \,{x^3}\] we should differentiate it using chain rule.
So as we know earlier, the equation of chain rule is defined as following:
\[ \dfrac{d}{{dx}}(f(g(x)))\, = {f'}(g(x)).{g'}(x) - - - - (A) \\
  f(g(x)) = \cos ({x^3})\, \Rightarrow {f'}(g(x)) = \, - \sin ({x^3}) \]
and \[\,g(x) = {x^3} \Rightarrow {g'}(x) = 3{x^2} \\ \]
Here, chain rule is used because it is a composite function. Here we use composite function rules to find the derivative of \[\cos \,{x^3}\].
Substitute these values in (A) and we will get as:
\[ \Rightarrow \dfrac{d}{{dx}}(\cos \,{x^3})\, = - \sin \,{x^3}.3{x^2}\, = - 3{x^2}\sin {x^3}\]

And thus we found the derivative of \[\cos \,{x^3}\]as\[ - 3{x^2}\sin {x^3}\].
Additional information:
The chain rule works with composites with more than two functions. When calculating the derivative of a composite of more than two functions, keep in mind that the composite of f, g, and h (in that order) is the composite of f with \[g\, \circ \,h\]. To compute the derivative of \[f \circ g \circ h\], the chain rule states that computing the derivative of f and the derivative of \[g\, \circ \,h\] is necessary.

Note: The chain rule is a method for determining the derivative of composite functions, with the number of functions in the composition determining the number of differentiation steps required.
\[\dfrac{d}{{dx}}(f(g(x)))\, = {f'}(g(x)).{g'}(x)\]
Since the composite function f is made up of two functions, g and h, you must differentiate \[f(x)\] using the derivatives \[{g'}\] and \[{h'}\].