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Chapter 3: Problem 4

State the derivative rule for the exponential function \(f(x)=b^{x}\) How does it differ from the derivative formula for \(e^{x} ?\)

Short Answer

Expert verified
Answer: The derivative formula for \(f(x) = b^x\) is given by \(\frac{d}{dx}(b^x) = b^x \cdot (\ln b)\), whereas the derivative formula for \(f(x) = e^x\) is \(\frac{d}{dx}(e^x) = e^x\). The difference between the two derivative formulas is the presence of \(\ln b\) in the formula for \(b^x\). In the case of \(e^x\), since \(b = e\), \(\ln b = \ln e = 1\), so the derivative is equal to \(e^x\).

Step by step solution

01

Derivative rule for \(f(x)=b^x\)

To find the derivative of \(f(x)=b^x\), we can use the chain rule: $$\frac{d}{dx}(b^x) = \frac{d}{dx}(e^{\ln b^x})$$ Since \(\ln b^x=x\ln b\), we can rewrite this as: $$\frac{d}{dx}(e^{\ln b^x}) = \frac{d}{dx}(e^{x\ln b})$$
02

Apply the chain rule

Now we can apply the chain rule to find the derivative: $$\frac{d}{dx}(e^{x\ln b}) = e^{x\ln b} \cdot \frac{d}{dx}(x\ln b)$$ Since \(\frac{d}{dx}(x\ln b) = \ln b\), we can rewrite this as: $$\frac{d}{dx}(e^{x\ln b})=e^{x\ln b}\cdot (\ln b) $$
03

Write derivative rule in terms of \(b^x\)

We can now write the derivative rule in terms of \(b^x\): $$\frac{d}{dx}(b^x)= b^x\cdot (\ln b) $$
04

Derivative formula for \(e^x\)

The derivative formula for \(e^x\) is simply: $$\frac{d}{dx}(e^x) = e^x$$
05

Compare the two derivative formulas

Comparing the two derivative formulas, we see that for \(f(x)=b^x\), the derivative is given by: $$\frac{d}{dx}(b^x)= b^x\cdot (\ln b) $$ Whereas for \(e^x\), the derivative is given by: $$\frac{d}{dx}(e^x) = e^x$$ The difference between the derivative formula for \(b^x\) and the derivative formula for \(e^x\) is the presence of \(\ln b\) in the formula for \(b^x\). For \(e^x\), since \(b=e\), \(\ln b = \ln e = 1\), so the derivative is equal to \(e^x\).

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