Question

How do you find the derivative of ${{e}^{-6x}}+e$?

Answer 1

To solve the given question, we should know the derivatives of some of the functions, and how to differentiate composite functions. The composite functions are functions of the form $f\left( g(x) \right)$, their derivative is found as, $\dfrac{d\left( f\left( g(x) \right) \right)}{dx}=\dfrac{d\left( f\left( g(x) \right) \right)}{d\left( g(x) \right)}\dfrac{d\left( g(x) \right)}{dx}$. The functions whose derivatives we should know are {{e}^{x}}$$$$,ax and constant function, their derivatives are ${{e}^{x}},a\And 0$ respectively. We will use these to find the derivative of the given function. We should also know that the derivative can be separated over addition.

Complete step by step solution:
We know that the derivative of the composite function is evaluated as $\dfrac{d\left( f\left( g(x) \right) \right)}{dx}=\dfrac{d\left( f\left( g(x) \right) \right)}{d\left( g(x) \right)}\dfrac{d\left( g(x) \right)}{dx}$.
We are given the function ${{e}^{-6x}}+e$, we are asked to find its derivative. As we know that the derivative can be separated over addition of function. So, we can evaluate its derivative as,
$\dfrac{d\left( {{e}^{-6x}}+e \right)}{dx}=\dfrac{d\left( {{e}^{-6x}} \right)}{dx}+\dfrac{d\left( e \right)}{dx}$
The ${{e}^{-6x}}$ is a composite function of the form $f\left( g(x) \right)$, here we have $f(x)={{e}^{x}}\And g(x)=-6x$.
To find the derivative of the given function, we need to find, $\dfrac{d\left( {{e}^{-6x}} \right)}{d\left( -6x \right)}\And \dfrac{d\left( -6x \right)}{dx}$.
We know that the derivative of ${{e}^{x}}$ with respect to x is ${{e}^{x}}$ itself. Thus, the derivative of ${{e}^{-6x}}$ with respect to $-6x$ must be equal to ${{e}^{-6x}}$. Hence, we get $\dfrac{d\left( {{e}^{-6x}} \right)}{d\left( -6x \right)}={{e}^{-6x}}$. Also, the derivative of $-6x$ with respect to x is 6.
The e is a constant function here, hence its derivative will be zero.
$\dfrac{d\left( {{e}^{-6x}}+e \right)}{dx}=\dfrac{d\left( {{e}^{-6x}} \right)}{dx}+\dfrac{d\left( e \right)}{dx}$
$\Rightarrow \dfrac{d\left( {{e}^{-6x}}+e \right)}{dx}=\dfrac{d\left( {{e}^{-6x}} \right)}{d\left( -6x \right)}\dfrac{d\left( -6x \right)}{dx}+0$
Substituting the expressions for the derivative, we get

& \Rightarrow \dfrac{d\left( {{e}^{-6x}}+e \right)}{dx}={{e}^{-6x}}\left( -6 \right) \\\ & \Rightarrow \dfrac{d\left( {{e}^{-6x}}+e \right)}{dx}=-6{{e}^{-6x}} \\\ \end{aligned}$$ **Thus, the derivative of the given function is $$-6{{e}^{-6x}}$$.** **Note:** The given function has terms of the form $${{e}^{ax}}$$. There is a special method to find the derivatives of these types of functions. We can find their derivative using the following method as: $$\dfrac{d\left( {{e}^{ax}} \right)}{dx}={{e}^{ax}}a$$ .