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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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Find $\lim\limits_{h\to 0}\frac{f(2-h^2)-f(2)}{h^2}$

let $f(x)= (x^3+2x)[\frac{x}{2}]$ [x]:floor function then : $$\lim\limits_{h\to 0}\frac{f(2-h^2)-f(2)}{h^2}=?$$ My try : $$\lim\limits_{h\to 0}\frac{f(2-h)-f(2)}{h}=f'(2)$$ $$\lim\limits_{x\to…
Almot1960
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$\lim\limits_{x \to a} \frac{f(x)-f(a)}{x-a}=L \in \mathbb{R} $

let function $f$ at $x=a\in \mathbb{R}$ be differentiable and $n ,m , k \in \mathbb{R}$ then prove that : $$\lim\limits_{h \to 0} \frac{f(a+mh)-f(a+nh)}{kh}=\frac{m-n}{k}f'(a) $$ My Try : since function $f$ at $x=a\in \mathbb{R}$ is …
Almot1960
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Differentiating $ x^{a}y^{b} = c $, in its simplest form.

$$ x^{a}y^{b} = c, $$ where a, b and c are constants. My attempts so far $$ \frac{dy}{dx} = ax^{a - 1}by^{b - 1}$$ $$ \frac{d^2y}{dx^2} = (a^2 - a)x^{a-2}(b^2 - b)y^{b - 2} $$ I think that these first and second derivatives are correct, however my…
John Allen
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If $x = (a^{\sin^{-1}t})^{1/2}$ and $y = (a^{\cos^{-1}t})^{1/2}$, show that $\frac{dy}{dx} = \frac{-y}x$

If $x = (a^{\sin^{-1}t})^{1/2}$ and $y = (a^{\cos^{-1}t})^{1/2}$, show that $\frac{dy}{dx} = -\frac{y}x$ I have tried solving this sum and led to nowhere, I am not able to eliminate a from this equation. Thanks in advance
Dhruv Raghunath
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Prove a function with 2 variables is not differentiable

$$f(x,y)={xy^2\over x^2+y^2}, \text{ with } f(0,0)=0$$ Show that $f$ is not differentiable at $(0,0)$. I know that by definition, I need to show that $$\lim_{h->0} \frac{||f(x+h,y+h)-f(0,0)||}{||h||}=0$$ (both partial derivatives are 0). However…
Annie Fannie
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nature of turning point of $\frac{x}{e^x-1}$ at $0$

The function $\frac{x}{e^x-1}$ has limit $1$ at $0$. It has derivative defined everywhere except at the origin. At the origin, the derivative exists and it is $0$. The limit of the second derivative is positive, but it is not a minimum point. It is…
Lost1
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Derivative of x[x]

What will be the derivative of x[x] when x is not an integer? I was applying the product rule but I'm stuck in the part were I have to differentiate [x] wrt x. [.] Denotes greatest integer function
user423865
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Using the definition of the derivative, prove that g(x) is not differentiable at x = 0.

Using the definition of the derivative, you are required to prove that $$g(x) = \sqrt{\frac{x^2(1+x)}{1-x}}, -1\le x \lt1$$ is NOT differentiable at x = 0. I know there exists a cusp at x = 0, but I need to use the definition of the derivative. Also…
chris24
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Differentiate $xx^T$ wrt to $x \in \mathbb{R}^n$

The problem is to find gradient and hessian of: $f(x) = \frac 1 2 || xx^T -A||_F^2$ given $A \in \mathbb{R^{n \times n}}$, $A$ is symmetric, $x \in \mathbb{R}^n$. Notation: $||\cdot||_F$ is a Frobenius norm of a matrix. My progress with problem…
Dmitri K
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Can you tell me the flaw in my differentiation of $e^{x^x}$?

I was killing time in a meeting where I wasn't needed and tried to calculate $\tfrac{\mathrm{d}}{\mathrm{d}x}\left[e^{x^x}\right]$. I already know I'm wrong from looking up the answer, but I'd like to know where I went amiss, if you had a minute.…
Charlie
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When computing the derivative of $x^2 + y^2$, why does $y$ become $0$?

The derivative of $$x^2 + y^2$$ is $$2x$$ I figured it out by using the calculator. Why does $y$ become $0$? Do I always think $y$ as $0$ in that situation?
tom
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find extremum of $y = x(x-1)^{\frac{1}{3}}$

given function: $$y = x(x-1)^{\frac{1}{3}}$$ steps: $$y'=\frac{(x^2-x)^{\frac{-2}{3}}2x-1}{3}$$ After simplifying: $$ y' = \frac{2x-1}{(3x^2-3x)^{\frac{2}{3}}}$$ therefore $$ x = \frac{1}{2}$$ $$x \ne 0$$ Am I right with calculus?
M.Mass
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derivative in higher dimension

Let $ f: R^{n} \rightarrow R^{p}$ and $ g:R^{n} \rightarrow R^{q}$ be differentiable functions. Define $ B(f,g)(a)=(f(a),g(a)) $, show that $ B(f,g) $ is also differentiable and $DB(f,g)a .(h) =B(Df(a)(h),g(a))+B(f(a),Dg(a)(h)) $. $$ $$ To answer…
Arifa zaman
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Prove that $\frac{x^2}{2} > x\cos x - \sin x$ for every $x\neq0$

I'm trying to prove that $\frac{x^2}{2} > x\cos x - \sin x$ for every $x\neq0$ $f(x) =\frac{x^2}{2} - x\cos x + \sin x $. I need to prove that $f > 0 $ for every $x\neq 0$. $f(0) = 0$. $f'(x) = x(1+\sin x)$ for every $x > 0$, $ f'(x) \geq 0$ and…
user371583
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Find the derivative of $f(x)=7x\ln |x|$

Find the derivative of $$f(x)=7x\ln|x|$$ How do they get the answer $$f'(x)= 7\ln|x|+7$$
Fiona Lu
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