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Question: What is the derivative of \({e^{ - 2x}}\)?...

What is the derivative of e−2x{e^{ - 2x}}?

Explanation

Solution

We will use the derivative of the exponential function formula to find the derivative in the first step then later we will also use the chain rule of the composite function. The derivative of the exponential function is the exponential function itself and then will take the derivative of its power.

Complete step by step answer:
Take the given expression: e−2x{e^{ - 2x}}
Apply the derivative of the given exponential function which is the exponential function by using the formula ddx(ex)=ex\dfrac{d}{{dx}}\left( {{e^x}} \right) = {e^x}and then apply chain rule for its power.
ddx(e−2x)=e−2xddx(−2x)\dfrac{d}{{dx}}\left( {{e^{ - 2x}}} \right) = {e^{ - 2x}}\dfrac{d}{{dx}}( - 2x)
Take constant outside the bracket along with the negative sign.
ddx(e−2x)=−2e−2xddx(x)\dfrac{d}{{dx}}\left( {{e^{ - 2x}}} \right) = - 2{e^{ - 2x}}\dfrac{d}{{dx}}(x)
Apply the formula - d(xn)dx=n.xn−1\dfrac{{d({x^n})}}{{dx}} = n.{x^{n - 1}}
ddx(e−2x)=−2e−2x1.x1−1\dfrac{d}{{dx}}\left( {{e^{ - 2x}}} \right) = - 2{e^{ - 2x}}1.{x^{1 - 1}}
Simplify the above expression finding the difference of the powers.
ddx(e−2x)=−2e−2x1.x0\dfrac{d}{{dx}}\left( {{e^{ - 2x}}} \right) = - 2{e^{ - 2x}}1.{x^0}
Anything raised to zero is equal to one.
ddx(e−2x)=−2e−2x\dfrac{d}{{dx}}\left( {{e^{ - 2x}}} \right) = - 2{e^{ - 2x}}
This is the required solution.
∴\therefore The derivative ddx(e−2x)=−2e−2x\dfrac{d}{{dx}}\left( {{e^{ - 2x}}} \right) = - 2{e^{ - 2x}}.

Additional Information:
In mathematics, integration is defined as the concept of calculus and it is the act of finding the integrals. One can find the instantaneous rate of change of the function at a point by finding the derivative of that function and placing it in the x-value of the point. Instantaneous rate of change of the function can be denoted by the slope of the line, which states how much the function is increasing or decreasing as the x-values change.

Note: Always Know the difference between the differentiation and the integration and apply its formula accordingly. Differentiation can be symbolized as the rate of change of the function, whereas integration signifies the sum of the function over the range. Integration and derivation are inverses of each other.