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Chapter 3: Problem 37

Use the General Power Rule where appropriate to find the derivative of the following functions. $$g(y)=e^{y} \cdot y^{e}$$

Short Answer

Expert verified
Answer: The derivative of the function \(g(y)=e^y\cdot y^e\) is \(g'(y) = e^y(y^e + e\cdot y^{e-1})\).

Step by step solution

01

Find the derivative of \(u\)

To differentiate \(u=e^y\) with respect to \(y\), we should recall that the derivative of \(e^y\) is simply itself. Thus, $$u'(y) = \frac{d}{dy}(e^y) = e^y.$$
02

Find the derivative of \(v\)

Now, we need to differentiate \(v=y^e\) with respect to \(y\). Since \(e\) is a constant, we can use the power rule, which states that the derivative of \(y^n\) is \(n\cdot y^{n-1}\). In this case, we have $$v'(y) = \frac{d}{dy}(y^e) = e\cdot y^{e-1}.$$
03

Apply the General Power Rule

Now that we have \(u'(y)\) and \(v'(y)\), we can apply the General Power Rule to find the derivative of the product of \(u\) and \(v\): $$g'(y) = u'(y) \cdot v(y) + u(y) \cdot v'(y).$$ Plugging in the expressions we found earlier for \(u'(y)\), \(v(y)\), \(u(y)\), and \(v'(y)\), we get $$g'(y) = (e^y) \cdot (y^e) + (e^y) \cdot (e\cdot y^{e-1}).$$
04

Simplify the result

We can simplify this expression a bit by factoring out the common factor of \(e^y\) from both terms: $$g'(y) = e^y(y^e + e\cdot y^{e-1}).$$ So, the derivative of the given function \(g(y)=e^y\cdot y^e\) is $$g'(y) = e^y(y^e + e\cdot y^{e-1}).$$

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Most popular questions from this chapter

Find the derivative of the inverse of the following functions at the specified point on the graph of the inverse function. You do not need to find \(f^{-1}\) $$f(x)=x^{2}+1, \text { for } x \geq 0 ;(5,2)$$

Compute the derivative of the following functions. \(h(x)=\frac{(x+1)}{x^{2} e^{x}}\)

Calculate the following derivatives using the Product Rule. $$\begin{array}{lll} \text { a. } \frac{d}{d x}\left(\sin ^{2} x\right) & \text { b. } \frac{d}{d x}\left(\sin ^{3} x\right) & \text { c. } \frac{d}{d x}\left(\sin ^{4} x\right) \end{array}$$ d. Based upon your answers to parts (a)-(c), make a conjecture about \(\frac{d}{d x}\left(\sin ^{n} x\right),\) where \(n\) is a positive integer. Then prove the result by induction.

Product Rule for three functions Assume that \(f, g,\) and \(h\) are differentiable at \(x\) a. Use the Product Rule (twice) to find a formula for \(\frac{d}{d x}(f(x) g(x) h(x))\) b. Use the formula in (a) to find \(\frac{d}{d x}\left(e^{2 x}(x-1)(x+3)\right)\)

Proof of \(\frac{d}{d x}(\cos x)=-\sin x\) Use the definition of the derivative and the trigonometric identity $$ \cos (x+h)=\cos x \cos h-\sin x \sin h $$ to prove that \(\frac{d}{d x}(\cos x)=-\sin x\)
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