What is the derivative of e^cx ?
if y = e^(cx)
the dy/dx = c(e^(cx)) assuming c is a constant.
Ah, the derivative of e^cx, you're asking about the "expo"! Well, get ready for some mathematical amusement. To find the derivative of e^cx, we can apply some clown tricks. Have you ever seen a clown multiply by itself over and over? That's what's happening here!
Now, let's get down to business. The derivative of e^cx is simply c * e^cx. Why, you ask? Well, think of it this way: e^cx is like a clown juggling balloons. When you take the derivative, it's like adding that extra c in front, as if the clown found a way to juggle more balloons. So, the derivative of e^cx is c * e^cx, just like a clown making balloon animals of different shapes and sizes!
I hope that made you smile, or at least helped you understand the concept! Remember, math can be funny too!
To find the derivative of e^cx, where c is a constant, you can use the chain rule.
Let y = e^cx. Taking the natural logarithm of both sides, we get ln(y) = cx.
Now, we can implicitly differentiate both sides with respect to x:
d/dx (ln(y)) = d/dx (cx)
To find the derivative of ln(y), we'll use the chain rule again:
1/y * dy/dx = c
Now, replace y with its original expression:
1/e^cx * dy/dx = c
Multiply both sides by e^cx:
dy/dx = c * e^cx
Hence, the derivative of e^cx with respect to x is c * e^cx.
To find the derivative of the function f(x) = e^(cx), you can use the chain rule. The chain rule states that if you have a composite function f(g(x)), the derivative of f(g(x)) is given by f'(g(x)) * g'(x).
Let's apply the chain rule to our function f(x) = e^(cx). We can let g(x) = cx, so that f(x) = e^g(x). The derivative of g(x) is g'(x) = c.
Now, we need to find the derivative of f(x) with respect to x, which is f'(x). Using the chain rule, we get:
f'(x) = f'(g(x)) * g'(x)
Since f(x) = e^g(x), we can write this as:
f'(x) = e^(g(x)) * g'(x)
Substituting the values of g(x) and g'(x), we have:
f'(x) = e^(cx) * c
Therefore, the derivative of f(x) = e^(cx) with respect to x is f'(x) = c * e^(cx).