3. Find the derivative of the function. Simplify where...

3. Find the derivative of the function. Simplify where possible. \begin{align*} (a) \ y &= \frac{e^x}{x^2} \\ (b) \ y &= \cos^{-1}(e^{2x}) \\ (c) \ f(x) &= \tan^{-1}(\ln x) \\ (d) \ y &= 3^{x \ln x} \\ (e) \ y &= \cot^{-1}t + \cot^{-1}\frac{1}{t} \\ (f) \ f(x) &= \sqrt{1-x^2} \arccos x \end{align*}

Transcript

00:01 Okay, so in this question we're asked to find a series of derivatives to certain functions.

00:06 So just to recall some of the rules, so here we have a product, right, of a function, let's call it u times a function v.

00:16 So the rule says that, let me write it into a solution, the derivative of f will be u prime times v plus u times v prime, derivative of the first times the second plus the first times the derivative of the second.

00:32 So derivative of x cubed, it's a power, so the three comes down and we reduce the power, and then we keep v, plus we keep u, and we do the derivative of cosine.

00:43 So derivative of cosine is minus sine, and then you need to multiply by the derivative of what's inside.

00:50 Derivative of three x is just three.

00:53 So this becomes three x squared cosine of three x minus three x cubed sine of three x.

01:03 And this is the final solution.

01:07 Now here, what we have to recall, let's again call this function u and this function v.

01:13 So now, when we have a quotient, the derivative of a quotient is derivative of the top one times the bottom one minus derivative of the bottom one times the top one over the bottom squared.

01:27 So let's make these calculations.

01:29 Derivative of the exponential is the exponential itself times the derivative of what's on top times three, and this multiplies by v x cubed plus x plus two minus the top one times the derivative of the bottom.

01:43 X cubed, three comes down and becomes squared.

01:46 X derivative is one, derivative of a constant is zero.

01:52 And this is all over x cubed plus x plus two, all squared.

01:57 So let's simplify on top...

Question

3. Find the derivative of the function. Simplify where possible. \begin{align} (a) \ y &= \frac{e^x}{x^2} \\ (b) \ y &= \cos^{-1}(e^{2x}) \\ (c) \ f(x) &= \tan^{-1}(\ln x) \\ (d) \ y &= 3^{x \ln x} \\ (e) \ y &= \cot^{-1}t + \cot^{-1}\frac{1}{t} \\ (f) \ f(x) &= \sqrt{1-x^2} \arccos x \end{align}

Question

3. Find the derivative of the function. Simplify where possible. \begin{align*} (a) \ y &= \frac{e^x}{x^2} \\ (b) \ y &= \cos^{-1}(e^{2x}) \\ (c) \ f(x) &= \tan^{-1}(\ln x) \\ (d) \ y &= 3^{x \ln x} \\ (e) \ y &= \cot^{-1}t + \cot^{-1}\frac{1}{t} \\ (f) \ f(x) &= \sqrt{1-x^2} \arccos x \end{align*}

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3. Find the derivative of the function. Simplify where possible. \begin{align} (a) \ y &= \frac{e^x}{x^2} \\ (b) \ y &= \cos^{-1}(e^{2x}) \\ (c) \ f(x) &= \tan^{-1}(\ln x) \\ (d) \ y &= 3^{x \ln x} \\ (e) \ y &= \cot^{-1}t + \cot^{-1}\frac{1}{t} \\ (f) \ f(x) &= \sqrt{1-x^2} \arccos x \end{align}