Question

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

Answer 1

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:

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'}$$.