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Questions tagged [derivatives]

Questions on the evaluation of derivatives or problems involving derivatives (for example, use of the mean value theorem).

The derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value)

Derivative of a function has a very natural geometric and physical interpretation: it corresponds to slope of the tangent line and to instantaneous velocity. In applications, it usually describes the rate of change of a physical variable.

Basic techniques used for computing the derivative of a given function are

It is useful to know the derivatives of elementary functions. This tag is intended for questions on the evaluation of derivatives.

Derivatives may be generalized to functions of several real variables. In this generalization, the derivative is reinterpreted as a linear transformation whose graph is (after an appropriate translation) the best linear approximation to the graph of the original function. The Jacobian matrix is the matrix that represents this linear transformation with respect to the basis given by the choice of independent and dependent variables. It can be calculated in terms of the partial derivatives with respect to the independent variables. For a real-valued function of several variables, the Jacobian matrix reduces to the gradient vector.

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How do find the fourth derivative of $e^{x^2}$

So using chain rule for the first derivative: $$f' = 2x\cdot e^{x^2}$$ then using product rule for the $$f'' = 2x(2xe^{x^2}) + e^{x^2}\cdot 2$$ Is there an easy way to get the third? Does the second deriative simplify to: $$4x^2e^{x^2} + 2e^{x^2} =…
Jwan622
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Let $y=(f(u)+3x)^2$ and $u=x^3-2x$. If $f(4)=6$ and $\frac{dy}{dx}=18$ when $x=2$, find $f'(4)$

Let $y=(f(u)+3x)^2$ and $u=x^3-2x$. If $f(4)=6$ and $\frac{dy}{dx}=18$ when $x=2$, find $f'(4)$. I'm sorry if I'm asking this beginner question, but I'm really confused about how to solve this problem. I've tried to substitute $u=4$ and other…
Zein IF
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Differentiating with respect to a product of two variables

I want to differentiate a function $C(h,u)$ associated with Manning friction: $$ C(h,u) = h^{-1/3} u \lvert u \rvert $$ According to WolframAlpha, $\mathrm{d} C / \mathrm{d} (hu)$ is $$ \frac{\mathrm{d}}{\mathrm{d}(hu)} h^{-1/3}u\lvert u \rvert =…
hertzsprung
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2nd order implicit derivative

What would the 2nd order implicit derivative of $y^2=12x$ be? I get the first derivative is $2yy'=12$ but Wolfram gives the second as $y''=\frac{-3}{xy}$ and I don't understand how they get that.
Ian
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When the derivative is equal to 1?

Looking at reciprocal functions, they have 2 “turning” points, where the higher derivative values near vertical asymptotes turn into the generally lower ones approaching the end behavior asymptote, for example, taking $$y=\frac1x$$ the points are…
H Huang
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Understanding definite integrals with functions as bounds

I can't quite grasp the meaning behind definite integrals defined on two bounds, which appear as functions. For instance, $$\int_{x^2}^{\cos x}t^2dt$$ What is this notation telling me? What does it mean that the lower bound is $x^2$, and the upper…
Julia Kim
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Dual Numbers and Automatic Differentiation

I have the following equation. I would like to solve this at the point x = a, but using dual numbers. $$f\left(x\right) = \dfrac{1}{x} + \sin\left(\dfrac{1}{x}\right)$$ Now, the derivative of this function is below, and that is the function we'd…
Workhorse
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Proving that a minimum is unique

I want to prove that a function $f(x)$ has its unique minimum at $f(c)$. By that I mean, $f(c) < f(x)$ for all $x \neq c$. The strict inequality is obviously vital. My question is, does this follow automatically if $f'(x) = 0$ if and only if $x =…
Masum
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Differentiate $ e^{-[y/b]^a} $ respect to y?

I would like to ask that, when I do the differentiation $ \frac{\partial }{ \partial y} e^{-[y/b]^a}$, my answer is $a \times e^{-[y/b]^a} \times (-1/b) $. Is this correct? I am not sure whether $(-1/b)$ should be added. Thank you very much for…
Chen
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Functional derivative for the same function expressed in different coordinates

This question arises when I'm reading section "3.3.1 Minkowski Space" of page 16-17 of the following document: http://www-thphys.physics.ox.ac.uk/people/JohnCardy/qft/qftcomplete.pdf On page 17, they took a functional derivative of Z[J] with respect…
user441951
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Derivative of $d^2x/dy^2$

Using the quotient rule and the chain rule you get $$ \frac{d}{dy}\left(\frac{dx}{dy}\right)=\frac{d}{dy}\left(\frac{1}{\frac{dy}{dx}}\right) =\frac{d}{dx}\left(\frac{1}{\frac{dy}{dx}}\right)\times \frac{dx}{dy}\\$$ I don't understand how this came.…
Aladdin
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Find points of discontinuity of $\log\left|{\frac{x+2}{x+3}}\right|$

I want to find the points of discontinuity of the following function: $$f(x)=\log\left|{\frac{x+2}{x+3}}\right|$$ This is defined for $x\neq-2$ and $x\neq-3$. I proceed to find the first derivative: $$f'(x)=\frac{1}{x^2+5x+6}$$ This is as well…
Cesare
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If $x:[0,a]\to \Bbb{R}^n$ is differentiable, what is $\frac{d}{dt}\Vert x(t)\Vert^2?$

Let $\Vert\;\Vert$ be the Euclidean norm on $\Bbb{R}^n$. Let $x:[0,a]\to \Bbb{R}^n$ be differentiable. How do I define \begin{align}\frac{d}{dt}\Vert x(t)\Vert^2?\end{align} Please, I need help on this! A detailed answer would suit me. Thanks a lot!
Omojola Micheal
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Differentiate $\frac{x^3}{{(x-1)}^2}$

Find $\frac{d}{dx}\frac{x^3}{{(x-1)}^2}$ I start by finding the derivative of the denominator, since I have to use the chain rule. Thus, I make $u=x-1$ and $g=u^{-2}$. I find that $u'=1$ and $g'=-2u^{-3}$. I then multiply the two together and…
Pablo
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Are there continuous functions $f\colon\mathbb{R}^2\to\mathbb{R}$ such that f has no directional derivatives?

Can the non-differentiability of a function $f:R^n \to R$ always be proved by using directional derivative? leads to the question whether there is some (continuous) function $f\colon\mathbb{R}^2\to\mathbb{R}$ such that for all…
Jens Schwaiger
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