Question

What is the derivative of \[{e^5}\] ?

Answer 1

Hint: Here we need to differentiate the given problem with respect to x. We know that the differentiation of constant term is zero and differentiation of \[{x^n}\] is \[\dfrac{{d({x^n})}}{{dx}} = n.{x^{n - 1}}\] . Here we have exponential constant. We know that the approximate exponential value is 2.718.

Complete step by step solution:
Given,
 \[{e^5}\] .
Now differentiating it with respect to ‘x’ we have,
 \[\dfrac{d}{{dx}}\left( {{e^5}} \right) = 0\] .
This is because we know that exponential function \[e\] is constant and \[{e^5}\] also.
(Suppose let's say that they asked us to find the integral or antiderivative of \[{e^5}\] then the answer is not zero.
 \[\int {{e^5}.dx = {e^5}x + c} \], where ‘c’ is the integration constant)
So, the correct answer is “0”.

Note:
\[ \bullet \] Linear combination rule: The linearity law is very important to emphasize its nature with alternate notation. Symbolically it is specified as \[h'(x) = af'(x) + bg'(x)\]
\[ \bullet \] Quotient rule: The derivative of one function divided by other is found by quotient rule such as \[{\left[ {\dfrac{{f(x)}}{{g(x)}}} \right] ^1} = \dfrac{{g(x)f'(x) - f(x)g'(x)}}{{{{\left[ {g(x)} \right] }^2}}}\] .
\[ \bullet \] Product rule: When a derivative of a product of two function is to be found, then we use product rule that is \[\dfrac{{dy}}{{dx}} = u \times \dfrac{{dv}}{{dx}} + v \times \dfrac{{du}}{{dx}}\] .
\[ \bullet \] Chain rule: To find the derivative of composition function or function of a function, we use chain rule. That is \[fog'({x_0}) = [(f'og)({x_0})] g'({x_0})\] .