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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 the absolute min and max in the given intervals

$k(x) = e^{-\frac{x^2}{2}}$ on $[-1,2]$ I think the derivative of that is $ -x e^{-\frac{x^2}{2}}$. I don't know how to find zero from that equation.
KPN123
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Simplification of the derivative

I have this equation $y=x^{5x^3}$ by doing a log transformation we get, $log (y) = 5x^3 log (x)$ upon doing a differentiation w.r.t $(x)$, we get $$\frac{1}{y}\frac{dy}{dx} = 5x^3.\frac{1}{x} + log(x) . 15x^2 =>\frac{dy}{dx} =…
Kamal
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Need help to simplify the derivative

Can someone tell me what would be the output of this equation? $$\frac{d}{dx}[\cos^4(x)\cdot\cos (x^4)] = -4x^3\cdot\cos^4(x)\cdot\sin (x^4)+4\cos(x^4)\cdot\cos^3(x)\cdot\sin(x)$$ But am not getting the same answer, would like to know what would be…
Kamal
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What does $\frac{dg}{dx}$ mean?

What does $\frac{dg}{dx}$ mean? Specifically, I'm trying to solve$$ \frac{1}{3}\frac{dg}{dx}\frac{1}{1+g^2} $$ where $$ g(x) = \frac{3x\left(1-x^2\right)}{x^4-4x^2+1} $$ I know $\frac{d}{dx}$ just means differentiate with respect to $x$ but I…
chriskinda
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Simple differentiation issue on multivariate function $u(x,t)$

So I have a function $u:\mathbb{R} \times (0,\infty) \to \mathbb{R} $ and a constant $a \in \mathbb{R}.$ Define $v:\mathbb{R} \times (0,\infty) \to \mathbb{R}$ by $v(x,t)=u(x+at,t)$. What is $\frac{\partial}{\partial t}v(x,t)$ in terms of $u$? Is it…
Tom Offer
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How to calculate this derivative?

I have just seen this notation of a question: Find $$\frac{d(x-x\sin(x))}{d(1-\cos(x))}$$ or something along those lines. I am well aware of notation like $\frac{dy}{dx}$ or something like $\frac{d(\sin(x))}{dx}$ but I don't really know what the…
Johannes
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How do you derive this easy to find the max/min points

How do you derive this easy to find the max/min points (There aren't actually any stationary points) $$ \dfrac {-24 x^2 -88 x -18} {16 x^2 +64 x +16} $$ I know how to use the quotient rule, but I think this has to be simplified at first, else it…
user163990
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Investigating monotone and bounded nature of a function.

If $$f(x)=x^3+bx^2+cx+d$$ and $0
jeevith
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monotonicity by examining the sign of derivative

given : $x>0 ,y >0 ,b>a>0$ prove the following by using derivative of a appropriate function: $${(x^b+y^b)}^{(1/b)} < {(x^a+y^a)}^{(1/a)}$$ I tried using $f(x)=(m^x+n^x)^{(1/x)}$ and $f(x)={(1+k^x)}^{(1/x)}$
user143997
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Complex Analysis using derivatives

I have been studying Euler's Formula and its derivation. In an article I am reading, I came across a use of derivatives I did not understand and am hoping someone can explain it. The use of derivatives is in this article. Let $z= a+bi$ for $e^z$…
Elliott
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What is the smallest possible value of their sum?

The product of two positive numbers is 36. What is the smallest possible value of their sum? so far I got $$xy=36$$ $$y=\frac{36}{x}$$
Gunz
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How to find where the function is decreasing/increasing/concave/convex $f(x) ={\frac{2}{1+x^2}}$?

$f(x) ={\frac{2}{1+x^2}}$ I need to find where this function is increasing, decreasing, concave and convex. I've found it's derivative: $f'(x)=\frac{-4x}{(1+x^2)^2}$ Now you're supposed to make either $f'(x)>0$ when it's increasing and $f'(x)<0$…
peroxy
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Is this function differentiable in $(0,0)$

Consider the function: $$f:\mathbb{R^2}\rightarrow\mathbb{R}$$ $$f(x,y)=\frac{x^2y^2}{x^4+y^2}\forall (x,y)\neq(0,0)$$ $$f(0,0)=0$$ It's clearly differentiable for all $(x,y)\neq(0,0)$. I have shown that both partial derivatives at the origin are…
F.Webber
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Gateaux and Frechet differentiability

Please help me to investigate Gateaux and Frechet differentiability of the functional $x \rightarrow ||x||_c$ depending on $x \in c$. The same about functionals $x \rightarrow ||x||_{c_0},\ x \in c_0$ and $x \rightarrow ||x||_{L_1[0,1]},\ x \in…
bga14
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Nth Derivative of the function

Find the $n^{th}$ derivative of $$f(x) = e^x\cdot x^m$$ If i am not wrong i have following $1^{st}$ Derivative: $e^x\cdot m \cdot x^{m-1} + x^m\cdot e^x$ $2^\text{nd}$ Derivative: $e^x\cdot m \cdot (m-1)\cdot x^{m-2} + 2 \cdot e^x\cdot m \cdot…
rndflas
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