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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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Differentiability of the function $ f(x, y) = | e ^x-y | (e^x-1) $

Prove that the function $ f: \mathbb R^2 \to \mathbb R $ defined by the formula $ f(x, y) = | e ^x-y | (e^x-1) $ is differentiable at $ (a, b ) \in \mathbb R ^ 2 $ if and only if $ e ^ a \neq b $ or $ a = 0, b = 1 $. I know that function is…
john1235
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Derivative of $\tan^{-1} \sqrt{\frac{1-\cos x}{1-\sin x}}$

Is there any way to differentiate $\tan^{-1} \sqrt{\frac{1-\cos x}{1-\sin x}}$ without any messy calculations? Here are my thoughts: We write $\cos x$ and $\sin x$ in terms of $\tan{\frac{x}{2}}$. Then our function becomes $\tan^{-1}…
madness
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Is this method of finding the derivative of $x^x$ acceptable in math?

I wanted to find the derivative of the $x^x$, and I came up with the following, and I want to know if it's acceptable to do so: Let's put $x^x = y$ $x^x = y $ $\ln(x^x) = \ln(y)$ $x\ln(x) = \ln(y)$ $d[x\ln(x)]/dx = [\ln(y)']/dx$ $\ln(x) + 1 =…
Rayan
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Problem on Mean Value Theorem

The Question: Let $f$ be a function defined on an interval $[a,b]$. What conditions could you place on $f$ to guarantee that $$ \min f'\leq \frac{f(b)-f(a)}{b-a} \leq \max f' $$ My Answer We must require a $\min f'$ and a $\max f'$ before we can…
Abhishek A Udupa
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If $p^2=a^2(\cos x)^2+b^2(\sin x)^2$, prove that $p +\frac{d^2 p}{dx^2}=\frac{a^2 b^2}{p^3}$

How to prove this? It seems simple enough to start but the at the end I cannot prove the expression $p +\frac{d^2 p}{dx^2}=\frac{a^2 b^2}{p^3}$. Is there a particular trick I am missing or is this just an ordinary sum with lots of manipulation. I…
Afnan Shah Qureshi
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How to find the maximum distance between two paths over a time interval (diff eqn) AP Calc Question

Alice and Bob go for a jog in the same direction along a straight path. For $0\leq t \leq20$, Alice’s velocity at time t is given by $A(t)=\frac{6010}{t^2-3t+50.5}$ meters per minute, and Bob’s velocity at time t is given by $B(t)=8.5t^3e^{-0.45t}$…
user130306
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Proving $f(x) = x^2 \sin(1/x)$, $f(0)=0$ is differentiable at $0$, with derivative $f'(0)= 0$ at zero

I need a solution for this question. I've been trying out this question for days and I haven't been able to find out its solution yet. And some explanation would help too. Show that the function f defined by: $$f(x):= \begin{cases} x^2\sin(1/x)…
Sreeraj Rajan
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Derivative of Function with Exponentials

I would like to know the derivatives of the following function: $ y = x^e*e^x$ At first sight it looks like the product rule should be used and so one would get $e*x^{e-1}*e^x+x^e*e^x$. Is this correct?
user66280
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Question about derivative

I need to check whether I've done it correctly To find whether a point is maximum of function $f(x)$, we have to checked whether $f''(x)>0, f''(x)=0$ or $f''(x)<0?$ To find the inflection point of the function, we have to find, $f''(x)=0, f'(x)=0,$…
user2041143
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How to get the derivative based on the function?

Based on $f(x) = x^2$, we know $f'(x) = 2x$. If $x = y^2$ Then $f(y) = (y^2)^2 = y^4$, and $f'(y) = 4y^3$. My question is if we only know $f'(x) = 2x$ $x = y^2$ How to get $f'(y) = 4y^3$ ? Any hint or formula will be helpful. Thanks
Hongbo Miao
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derivative of zero order bessel function of first kind

I want to know the derivative of the zero order bessel function of first kind ($J_0(x)$). and how it changes with $x$ and I also would like to know the roots of this function. could anyone help?
zahra
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$7$ th derivative of function at $x=0$

If $\displaystyle y=\frac{\sin(x^2)-x^2}{x^3}$. Then value of $\displaystyle \frac{d^7y}{dx^7}\bigg|_{x=0}=$ What i try $$\sin x=\sum^{\infty}_{n=0}(-1)^n\frac{x^{2n+1}}{(2n+1)!}$$ Replace $x\rightarrow x^2$ Then $$\sin…
jacky
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Trying to find the lyapunov function

I have the system that I want to show the global asymptotic stability of the origin $$\dot{x_1} = x_2 \\ \dot{x_2} = -g(k_1 x_1 + k_2 x _2) $$ where k1 and k2 are positive numbers. Also, $$g(y)y > 0 $$ for all $y\neq 0$ and $$\lim_{|y|\rightarrow…
user1426485
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Is the nth time derivative of position ever really zero?

Change in velocity comes from acceleration, but that acceleration was also a leap that started from nothing. It was zero, then positive, and then something else. So it's application must have been pushed by some other, more interior, and…
user739960
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Two values of minima of a function by two methods.

I had a problem of finding minima of a function $$f(x)=2^{x^2}-1+\frac{2}{2^{x^2}+1}$$ I solved it using AM-GM inequality, $$2^{x^2}-1+\frac{2}{2^{x^2}+1}$$ $$2^{x^2}+1+\frac{2}{2^{x^2}+1}-2$$ $$2^{x^2}+1+\frac{2}{2^{x^2}+1}\ge\…
pranjal verma
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