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Chapter 3: Problem 5

What is the derivative of \(y=e^{k x} ?\) For what values of \(k\) does this rule apply?

Short Answer

Expert verified
Answer: The derivative of the function \(y = e^{kx}\) is \(\frac{d}{dx}(y) = k e^{kx}\). This rule applies for all real values of \(k\).

Step by step solution

01

Apply the Chain Rule

Since we have a composite function, \(y = e^{kx}\), we need to apply the chain rule to find its derivative. The chain rule states that if we have a function \(y = u(v(x))\), then the derivative \(y' = u'(v(x)) \cdot v'(x)\). Here, we let \(u(x) = e^x\) and \(v(x) = kx\).
02

Find the Derivatives of u and v

Now, we need to find the derivatives of \(u(x)\) and \(v(x)\). Derivative of \(u(x) = e^x\): $$u'(x) = \frac{d}{dx}(e^x) = e^x.$$ Derivative of \(v(x) = kx\): $$v'(x) = \frac{d}{dx}(kx) = k.$$
03

Apply the Chain Rule

Now, we apply the chain rule formula mentioned in Step 1: $$\frac{d}{dx}(y) = u'(v(x)) \cdot v'(x).$$ Substitute the derivatives of \(u\) and \(v\) that we found in Step 2: $$\frac{d}{dx}(y) = e^{kx} \cdot k.$$
04

Describe the Valid Values for k

The derivative expression we found in Step 3 is valid for all real numbers because the exponential function \(e^{kx}\) is well-defined for all real numbers x and k. Therefore, the derivative rule applies for all values of \(k\). In summary, the derivative of \(y = e^{kx}\) is: $$\frac{d}{dx}(y) = k e^{kx},$$ and this rule applies for all real values of \(k\).

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