Question

If we have an expression as \[{e^x} + {e^y} = {e^{x + y}}\], Show that \[\dfrac{{dy}}{{dx}} = - {e^{y - x}}\].

Answer 1

Hint: In this question, we have to derive the required solution from the given particulars.
We have to differentiate the given equation concerning x. For that first, we need to use the differentiation formula of the exponential function then using the given expression in that and by simplifying and using the property of the exponential function we will get the required result.
Formula:
Differentiation formula:
\[\dfrac{d}{{dx}}\left( {{e^x}} \right) = {e^x}\]
Property of the exponential function:
\[{e^{x - y}} = \dfrac{{{e^x}}}{{{e^y}}}\]

Complete step-by-step solution:
It is given that, \[{e^x} + {e^y} = {e^{x + y}}\]…(i)
We need to derive \[\dfrac{{dy}}{{dx}} = - {e^{y - x}}\] from the given equation.
Differentiating (i) using the differentiation formula concerning x we get,
\[\dfrac{{d{e^x}}}{{dx}} + \dfrac{{d{e^y}}}{{dx}} = \dfrac{{d\left( {{e^{x + y}}} \right)}}{{dx}}\]
\[\Rightarrow {e^x} + {e^y}\dfrac{{dy}}{{dx}} = {e^{x + y}}\left( {1 + \dfrac{{dy}}{{dx}}} \right)\] [Using the formula, \[\dfrac{d}{{dx}}\left( {{e^x}} \right) = {e^x}\]]
\[\Rightarrow {e^x} + {e^y}\dfrac{{dy}}{{dx}} = \left( {{e^x} + {e^y}} \right)\left( {1 + \dfrac{{dy}}{{dx}}} \right)\][Using the given equation, \[{e^x} + {e^y} = {e^{x + y}}\]
Simplifying we get,
\[{e^x} + {e^y}\dfrac{{dy}}{{dx}} = {e^x} + {e^x}\dfrac{{dy}}{{dx}} + {e^y} + {e^y}\dfrac{{dy}}{{dx}}\]
\[\Rightarrow {e^x}\dfrac{{dy}}{{dx}} + {e^y} = 0\]
Taking the terms \[\dfrac{{dy}}{{dx}}\] on the left-hand side and the other terms on the right-hand side we get,
\[\Rightarrow {e^x}\dfrac{{dy}}{{dx}} = - {e^y}\]
\[\Rightarrow \dfrac{{dy}}{{dx}} = - \dfrac{{{e^y}}}{{{e^x}}}\]
Using the division property of the exponential function we get,
\[\Rightarrow \dfrac{{dy}}{{dx}} = - {e^{y - x}}\]
Hence, \[\dfrac{{dy}}{{dx}} = - {e^{y - x}}\] (Proved).

Note: The derivative of a function of a real variable measures the sensitivity to change of the function value concerning a change in its argument. Derivatives are a fundamental tool of calculus.
Derivative of a function \[y = f\left( x \right)\] can be written as \[\dfrac{{dy}}{{dx}}\] or \[f'\left( x \right)\].
e is the irrational number called Euler's number. It's the second most commonly known irrational number after Pi. Its value is \[2.71828...\]and it goes on.
The x is an exponent (in \[{e^x}\]) indicating how many times to multiply e by itself. The x is a variable. If x is\[2\], that means\[e \times e\]. If x is\[6\], that means\[e \times e \times e \times e \times e \times e\].