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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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With $u = x^3$ and $v=\arcsin(x)$, prove $\frac{du}{dv}=3\sqrt{u(u^{1/3}-u)}$

If $u=x^3$ and $v=\arcsin(x)$, prove that $$\frac{du}{dv}=3\sqrt{u(u^{1/3}-u)}$$ I have tried $$\frac{du}{dv}=\frac{\frac{du}{dx}}{\frac{dv}{dx}}$$ but I can't prove it.
user80551
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$(\nabla \vec a)\vec b = (\vec b \nabla)\vec a$?

Sometimes I see this expression in my physics lectures where the professor writes: $(\nabla \vec a)\vec b = (\vec b \nabla)\vec a$. I wasn't sure about it, and I tried to explicitly calculate each expression (using the components) and as I expected,…
imbAF
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Is the solution to picture on the left correct?

I know that h and h'' have to be one of those curves due to the concavity of f(x) at f''(x). However I am having difficulty understanding how am I supposed to know what is h and what is h''. I think h and h'' should be switched in picure on the…
SSSNIPD
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A function $f: \mathbb R \to \mathbb R$ satisfies equation $f(x+y)=f(x)f(y)$ for all $x,y\in \mathbb R $

Problem: A function $f: \mathbb R \to \mathbb R$ satisfies equation $f(x+y)=f(x)f(y)$ for all $x,y\in \mathbb R$ and $f(x)\neq 0$ for all $x \in \mathbb R$. If $f'(0)=2$, find $f'(x)$ in terms of $f(x)$ Solution: $f(x+y)=f(x)f(y)$ Differentiating…
rst
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Elementary question about derivative

i'm dealing with a really simple question that I'm struggling to find an answer because I'm messed up with physics notations. Suppose you have the relation $x=Ky$ and you have a function $f$. $K$ is a fixed scalar. You know explicitly $f$ as well…
Atmos
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Calculating $\frac{\partial^2 u}{\partial x^2}$

I'm having troubles calculating $\frac{\partial^2 u}{\partial x^2}$. I know that $$\frac{\partial u}{\partial x} = \cos \varphi \frac{\partial u}{\partial r}-\frac{\sin \varphi}{r} \frac{\partial u}{\partial \varphi}$$ $$\frac{\partial r}{\partial…
Quotenbanane
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Inconsistency in definitions of "critical points" and "differentiable"?

A student of mine is in a class that uses the following definition of derivative: $$f'(x)=\lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$$ Their definition of critical points is "places where the derivative of a function is either zero or undefined". Yet they…
user3925803
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Gradient of $\max_yf(x,y)$

Given $p(\textbf{x})$=$\max_{y}f(\textbf{x},y)$, do we have $\nabla p(\textbf{x})=\max_{y}(\nabla_{\textbf{x}}f(\textbf{x},y))$?
Kufscrow
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if the chain rule is f(g(x)) = g’(x)f’(g(x)) why does f(x+ au) gives gradient of f(x) with respect to x?

I dont understand the directional derivative of the function $f(x+ au)$ with respect to a when a =0. According to Goodfellow and al. We can see that, thanks to the chain rule, $\frac{d}{da }f(x+ au)$ evaluates to $u^T\nabla_xf(x)$ when a =0…
Revolucion for Monica
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Why monotonicity of the gradient implies the directional derivative is increasing

Suppose $f: \mathbb{R}^n \rightarrow \mathbb{R}$ is a differentiable function. Define $g(t)=f(x+t(y-x))$ for all $x, y \in \mathbb{R}^n$ and $t\geq 0$. Show when $\langle \nabla f(y) - \nabla f(x), y-x \rangle \geq 0$, $g'(t)=\langle \nabla f(x +…
Saeed
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Is it acceptable to factor out the "differentiand" from a derivative?

My physics professor likes to write the following: Let $D = \frac{\mathrm{d}}{\mathrm{d}x}$ therefore, $$ x - Dx = (1-D)x $$ This sort of makes my skin crawl; however, beyond the inelegance of the math, is there anything actually wrong about it?
Kalcifer
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Is the derivative both the best approximation of a value in relation to something else and how that value is changing with time?

For example is we have a function m(x) that describes the distribution of mass along a line, so m(x) gives you the amount of mass at a certain length of the line. If we take the derivative of m with respect to x we get dm/dx or density. So if I were…
CatsOnAir
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Second derivative notation?

The second derivative is defined as d/dx(dy/dx). I have multiplied so that it is simplified, so the second derivative is, d^2x/(dx)^2, yet in some books it states it as d^2x/d(x)^2. Can this limit not act as a fraction?
shreck357
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Help me understand first derivative in the context of distance and velocity

While studying math in high school, we learned a lot about derivatives and integrals, but we used handbooks of common derivatives, so we never got the chance to link derivatives to something from the real world. So I tried to understand them by…
Physther
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Find the derivative/ Differentiation

Find $f_y'(x,x^2)$ for a differentiable function $f(x,y)$ satisfying the conditions $f(x,x^2) = const, f_x'(x,x^2) = x$. I found that if $f(x,y) = \frac{1}{2}x^2 - \frac{1}{2}y$, it satisfies this conditions and $f'_y(x,x^2)$ would be $\frac{1}{2}$,…
Galep Impet
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