104 LearnersLast updated on September 1, 2025
Derivative of e^y
We use the derivative of e^y, which is e^y(dy/dx) when y is a function of x, as a tool to understand how the function changes with respect to a variable. Derivatives are crucial in various real-life applications including scientific modeling and economics. We will now discuss the derivative of e^y in detail.
What is the Derivative of e^y?
The derivative of ey is an important concept in calculus.
It is generally represented as d/dx(ey) or (ey)'.
Since e^y is an exponential function, its derivative is itself multiplied by the derivative of its exponent. The key concepts involved are:
Exponential Function: ey where e is a constant approximately equal to 2.71828.
Chain Rule: Used for differentiating composite functions like ey when y is a function of x. Implicit
Differentiation: Necessary when y is implicitly defined in terms of x.
Derivative of e^y Formula
The derivative of ey, when y is a function of x, can be denoted as d/dx(ey) or (ey)'.
The formula used to differentiate ey is: d/dx(ey) = ey(dy/dx)
This formula is applicable whenever y is a differentiable function of x.
Proofs of the Derivative of e^y
We can derive the derivative of e^y using different methods.
To show this, we will apply the chain rule and the concept of implicit differentiation.
Here are the methods used to prove this:
Using Chain Rule To prove the differentiation of ey using the chain rule, consider y as a function of x:
Let u = y, where y is a function of x.
Then, ey = eu.
Applying the chain rule, we get: d/dx(ey) = d/dx(eu) = eu * du/dx Since u = y, du/dx = dy/dx.
Therefore, d/dx(ey) = ey * dy/dx.
Using Implicit Differentiation Assume y is implicitly defined in terms of x.
Differentiate both sides of the equation ey = z with respect to x: d/dx(ey) = d/dx(z)
Using the chain rule, we get: ey * dy/dx = dz/dx
If z = e^y, then dz/dx = ey * dy/dx.
Higher-Order Derivatives of e^y
When a function is differentiated multiple times, the resulting derivatives are known as higher-order derivatives.
Higher-order derivatives can provide more insight into the behavior of the function.
Consider the case of ey:
First Derivative: d/dx(ey) = ey(dy/dx)
Second Derivative: Differentiate the first derivative again, using the product rule: d²/dx²(ey) = d/dx(ey(dy/dx)) = (ey(dy/dx))(dy/dx) + ey(d²y/dx²)
This process can be continued for higher-order derivatives.
Special Cases:
When y is constant, the derivative ey is simply ey multiplied by 0, which is 0. When y is a linear function of x, say y = ax, the derivative simplifies to ey * a.
Common Mistakes and How to Avoid Them in Derivatives of e^y
Students often make errors when differentiating ey, especially when y is a function of x. These mistakes can be corrected by understanding the correct process. Here are some common mistakes and how to solve them:
Mistake 1
Neglecting the Chain Rule
Students may forget to apply the chain rule, leading to incorrect results. Always remember to multiply by dy/dx when differentiating ey where y is a function of x.
Mistake 2
Ignoring Implicit Differentiation
Students sometimes overlook the need for implicit differentiation, especially when y is not explicitly given as a function of x. Make sure to account for dy/dx.
Mistake 3
Misapplying Product Rule
In cases where ey is part of a product, students might incorrectly apply the product rule. Ensure proper use of the product rule, differentiating each component correctly.
Mistake 4
Forgetting Constants in Exponential Functions
When dealing with expressions like e(y+z), students often forget to differentiate all variables involved. Remember to apply the chain rule to each term.
Mistake 5
Incorrectly Handling Higher-Order Derivatives
Mistakes occur when calculating higher-order derivatives. Ensure each differentiation step is performed accurately using applicable rules.
Examples Using the Derivative of e^y
Problem 1
Calculate the derivative of e^(2y) with respect to x, given y = x² + 1.
Here, we have f(y) = e(2y) and y = x² + 1.
Using the chain rule: d/dx(e(2y)) = e(2y) * d/dx(2y)
Since y = x² + 1, we have: d/dx(2y) = 2 * d/dx(y) = 2 * 2x = 4x
So, the derivative is: f'(x) = e(2y) * 4x = 4x * e(2(x²+1))
Thus, the derivative of e(2y) with respect to x is 4x * e(2(x²+1)).
Explanation
We differentiate e(2y) using the chain rule. By substituting y = x² + 1 and differentiating, we obtain the derivative.
Problem 2
A company observes growth modeled by e^y, where y is the profit function y = 3x + 2. Find the rate of change of growth at x = 1.
We have ey as the growth model and y = 3x + 2.
First, differentiate y: dy/dx = 3
Now, differentiate e^y: d/dx(ey) = ey * dy/dx = ey * 3
At x = 1, y = 3(1) + 2 = 5
Therefore, the rate of change of growth is: 3 * e5
Explanation
The company's growth is modeled by ey, where y is profit. The rate of change is found by differentiating and substituting x = 1 into the equation.
Problem 3
Find the second derivative of e^y where y = ln(x).
First, find the first derivative: d/dx(ey) = ey * dy/dx
Since y = ln(x), dy/dx = 1/x.
Therefore, d/dx(e^y) = ey * 1/x
Now, to find the second derivative: d²/dx²(ey) = d/dx(ey * 1/x)
Using the product rule: = (1/x) * d/dx(ey) + ey * d/dx(1/x) = (1/x) * (ey * 1/x) + ey * (-1/x²) = ey/x² - ey/x² = 0
So, the second derivative is 0.
Explanation
We start with the first derivative using implicit differentiation and apply the product rule for the second derivative. The terms simplify to zero.
Problem 4
Show that d/dx(e^(3y)) = 3e^(3y) * dy/dx.
Using the chain rule:
Consider f(x) = e(3y) d/dx(e(3y)) = e(3y) * d/dx(3y)
Since d/dx(3y) = 3 * dy/dx,
We have: d/dx(e(3y)) = 3e(3y) * dy/dx
Hence proved.
Explanation
We use the chain rule to differentiate e(3y). The derivative of the exponent is multiplied by the derivative of y, confirming the result.
Problem 5
Differentiate e^(y/x) with respect to x.
Use the chain rule and quotient rule:
d/dx(e(y/x)) = e(y/x) * d/dx(y/x)
Apply the quotient rule: d/dx(y/x) = (x * dy/dx - y * 1)/x² = (x * dy/dx - y)/x²
Therefore, the derivative is: e(y/x) * ((x * dy/dx - y)/x²)
Explanation
We apply the chain rule and quotient rule to find the derivative of e(y/x) by differentiating the exponential function and its exponent.
FAQs on the Derivative of e^y
1.What is the derivative of e^y?
2.How is the derivative of e^y used in real life?
3.Can we differentiate e^y when y is a constant?
4.What rule is used to differentiate e^(y/x)?
5.Are the derivatives of e^y and e^x the same?
Important Glossaries for the Derivative of e^y
- Derivative: A measure of how a function changes as its input changes, often represented as the slope of the tangent line to the curve of the function.
- Exponential Function: A function of the form ey where e is the base of the natural logarithm, approximately equal to 2.71828.
- Chain Rule: A fundamental rule in calculus used to differentiate compositions of functions.
- Implicit Differentiation: A method to find derivatives of functions that are not explicitly solved for one variable in terms of another.
- Quotient Rule: A technique for differentiating functions that are the ratio of two differentiable functions.
Jaskaran Singh Saluja
About the Author
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
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: He loves to play the quiz with kids through algebra to make kids love it.
