Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I am lost on deriving this one function, because im kind of confused with the $9e^x$ and the fraction part. Maybe if someone can guide me through the steps that would be awesome.

$$y=9e^x+\frac{2}{\sqrt[3]{x}}$$

share|improve this question
    
Is this homework? –  VF1 Oct 8 '12 at 23:58
add comment

3 Answers 3

Some rules that may help you:

For a function $f(x)$ and its derivative with respect to x $f'(x)$, $$ \frac{d}{dx} c f(x) = c f'(x) $$

You may want to look up what the derivative of the exponent function is. That's an important one to know. Also, for the fraction part, consider that: $$ \frac{2}{\sqrt[3]{x}} = \frac{2}{x^{1/3}} = 2 x^{-1/3} $$ For this one you may want to look up the power rule.

share|improve this answer
add comment

This is a rather cook-book answer, but. . .

As said above, use that $\frac{d}{dx}cf(x)=c\frac{d}{dx}f(x)$ and $\frac{2}{\sqrt[3]{x}}=2x^{-\frac{1}{3}}.$

Follow that up with the fact that $\frac{d}{dx}e^x=e^x$ and $\frac{d}{dx}x^m=mx^{m-1}.$

The first follows from many things (particularly the definition of $e^x$). The second follows from the definition of the derivative.

share|improve this answer
add comment

Just for instruction's sake, I'm going to solve this with logarithms. Bear in mind:

$$\log(a \cdot b) = \log a + \log b$$

and

$$\log(a^x) = x \log a$$

and

$$\log e = 1$$

Let $y_1 = 9e^x$ and $y_2 = 2x^{-1/3}$. Now,

$$\log y_1 = \log 9 + x$$

Differentiating w.r.t $x$,

$$\frac{1}{y_1} \frac{dy_1}{dx} = 1$$ Thus, $ \frac{dy_1}{dx} = y_1 = 9e^x$ (which is not surprising since that's a property of the exponential function).

Now, $\log y_2 = \log 2 -\frac{1}{3} \log x$. Hence, differentiating w.r.t $x$,

$$\frac{1}{y_2} \frac{dy_2}{dx} = -\frac{1}{3} \frac{1}{x}$$

Thus, $$\frac{dy_2}{dx} = -\frac{2}{3} x^{-4/3}$$

And therefore, $$\frac{dy}{dx} = 9e^x -\frac{2}{3} x^{-4/3}$$

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.