Mathematics Stack Exchange
Mathematics Stack Exchange

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.

33197 questions
2
votes
1 answer

Derivative of the composition of two differentiable functions

Calculate $$ \frac{\mathrm d}{\mathrm dt} f (g(t^2),g(t^4)), $$ where $f$ is a differentiable function of two variables and $g$ is a differentiable function of one variable. Your answer should be expressed in terms of $f, g$ and their derivatives…
Arsenalfan_179027
  • 21
2
votes
2 answers

The existence of second derivative

Let $f(x)=|x|x$ then $f''(0)$ does not exist. Why? If $x>0$, $f'(x)=2x$ and if $x<0$, $f'(x)=-2x$. Then when $x=0$, does $f'(x)$ also not exist?
email
  • 793
  • 2
  • 7
  • 10
2
votes
0 answers

Derivative of memory function

I'm working on an optimization problem that takes a huge amount of time to solve if I'm unable to compute the gradient of the objective function. However, I'm a bit unsure of how to proceed in these gradient calculations - I'm stuck in the…
Niclas
  • 21
2
votes
5 answers

Monotonicity of $f(x) =\sin(\ln(x))-\cos(\ln(x))$

Find the interval in which $f(x) =\sin(\ln(x))-\cos(\ln(x))$ is increasing. After differentiating we get $$f'(x) = \frac{\cos\left(\ln(x)\right)}{x} +\frac{\sin\left(\ln(x)\right)}{x}$$ Now how do we analyze this expression?
Pratyush Dutta
  • 23
2
votes
1 answer

How do I show that $V_n$ is as required?

Question : If $V_n=\frac{d^n}{dx^n}(x^n \log x)$, show that $V_n=nV_{n-1}+(n-1)!$ Hence show that $$V_n=n! (\log x + 1 + \frac{1}{2}+\frac{1}{3}+\dot{} \dot{} \dot{}+\frac{1}{n})$$ What I have managed to do so far: I have found that…
The Dead Legend
  • 4,727
2
votes
6 answers

Derivative of $x^2$

This seems too easy, but here's the question: $x^2$ is $x + x + ...+ x$ (with $x$ terms). Its derivative is $1 + 1 + ... + 1$ (also $x$ terms). So the derivative of $x^2$ seems to be $x$. And another expression: we know that if $y = nx$, then $y' =…
Tango
  • 29
2
votes
1 answer

Derivatives question involving tangent

Find the derivative of $2^{\tan(1/x)}$. I know that I should replace $\frac1x$ with $u$ and such, but then I can't continue it...
xdfg
  • 33
2
votes
2 answers

Prove $1/x^n$ is differentiable

Consider the function $g(x)=\frac{1}{x^n}$ where $n \in\Bbb N$, prove that g is differentiable. I tried to use the definition, Let $c \in\Bbb R$, then: $$\frac{g(x)-g(c)}{x-c}=\frac{\frac{1}{x^n}-\frac{1}{c^n}}{x-c}=\frac{c^n-x^n}{(x-c)(x^n \cdot…
Bob
  • 869
2
votes
3 answers

MCQ The nth derivative of $f(x)=\frac{1+x}{1-x}$

Let $f(x)=\dfrac{1+x}{1-x}$ The nth derivative of f is equal to: $\dfrac{2n}{(1-x)^{n+1}} $ $\dfrac{2(n!)}{(1-x)^{2n}} $ $\dfrac{2(n!)}{(1-x)^{n+1}} $ by Leibniz formula $$ {\displaystyle \left( \dfrac{1+x}{1-x}\right)^{(n)}=\sum…
Cyro
  • 61
2
votes
4 answers

Find Angle Between Two Curves at Point of Intersection

Find angle between these two curves at point of intersection : $$K_1: x^2y^2 + y^4 = 1$$ and $$K_2 : x^2 + y^2 = 4 $$ Thanks!
purplerain
  • 153
2
votes
5 answers

What is the derivative of $\frac{1}{f(x)g(x)}$?

I feel like this is a very basic question, yet I struggle with it immensely. I know that $(f(x)*g(x)) = f(x)'g(x)+f(x)g(x)'$, but how to use that in order to figure out $\frac{1}{f(x)g(x)}$ ?
Simplici7y
  • 123
2
votes
2 answers

Why doesn't derivative show the complex maxima minima?

Let's find the global minimum of $y=x^2$ First we calculate its first derivative, and make it equal to zero. $y'=2x=0$; $x=0$ Then we check its second derivative, if it's positive then it is minimum. $y''=2$ $(0,0)$ turned out to be global…
MCCCS
  • 1,625
2
votes
1 answer

Given distance covered as a function of time, find acceleration.

Since this question covers more aspects of differentiation and mathematical manipulation than kinematics, I am posting it here. My attempt : On differentiating position(x) once, we get velocity(v) and on differentiating position twice, we get…
Arishta
  • 948
2
votes
0 answers

Fractional derivative

It is mathematically, or symbollicly possible take the $n^{th}$ derivative of a function $f$ for values of n other than natural numbers? (or as I like to call it: literal partial derivatives). That is: $$\frac{d^nf}{dx^n}(x),n∉ℕ_1$$ Perhaps for…
Graviton
  • 4,462
2
votes
2 answers

Prove $f(x) =\ \frac{x^5}{5!} +\frac{x^4}{4!} +\frac{x^3}{3!}+\frac{x^2}{2!} +x+1$ has only one root.

We have to prove that the equation $\displaystyle \frac{x^5}{5!} +\frac{x^4}{4!} +\frac{x^3}{3!}+\frac{x^2}{2!} +x+1=0$ have exactly one real root . My sir told me it is just an application of derivative . But I could not understand what he mean by…
a25bedc5-3d09-41b8-82fb-ea6c353d75ae
  • 1,769
1 2 3
…
99 100