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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 intervals on which $f(x) = 8\cos 4(x)$ decreases for $0 \le x \le π $?

Find the intervals on which $f(x) = 8\cos 4(x)$ decreases for $0 \le x \le π $. What is the fast way to compute it?
FreeMind
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Differentiation with surds

I know dy / dx = n^n-1 I have the problem $y = \sqrt{x^2+2x}$ I have broken that down to $y = \sqrt{x^2 +2x}$. The x would differentiate to 1, but how do you differentiate the surd?
jordan b
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Derivative of simple fraction

What is the fastest way of finding the derivative of: $\frac{x}{x+K}$ (simplified form) is there a substitution I miss or does the quotient rule the job here? There should be a quick way of finding the derivative
Derk
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Compute $\frac{dy}{dx}$ as a function of $y$ when $x=\cos y$

The function $\cos^{-1}(x)$ is defined for $-1\leq x\leq1$ and $0\leq y\leq \pi$ by the equivalence $$y=\cos^{-1}(x)\Leftrightarrow x=\cos y$$ (a) Compute $\dfrac{dy}{dx}$ as a function of $y$ by differentiating the second equation in $(1)$. (b) Use…
Inverse Problem
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derivative of function of T

How do I take the derivative of: $\left ( \frac{1}{T^4} \right ) \left (\frac{1}{K-T} \right )$ Can I just use the product rule? IT seems like it get pretty complicated pretty fast
Jackson Hart
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When to use the chain rule

Would I use the chain rule in the following derivative problem: $$(sinx/x)$$ So far I have simplified it to: sin(x)(-1x^(-2))+x^(-1)(cosx) Would I have to further take the derivative of cosx Basically I do not quite understand when to use the chain…
Hello
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Derivatives with functions of two or more variables

For the function $ln(4x^2+4y^2)$ when taking the derivative with respect to $x$, do you essentially leave the $y$ terms alone? I received the answer of $\frac{2x+y^2}{x^2+y^2}$, however the books solution deems that to be wrong. Looking for some…
user153509
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Is there an easy way to study the sign of this?

I would like to study the sign of this derivate, but I don't know where to start : http://www.wolframalpha.com/input/?i=derivate+sqrt%28x%5E4-7x%5E2%2B16%29
Pop Flamingo
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demonstrate that function is increasing in intervals that are multiples of pi?

I have the derivative: $$- \frac1{x^2} + 1 + \frac{\cos^2(x)}{\sin^2(x)}$$ and am supposed to show that this is positive for all $x \in (n\pi, (n+1)\pi)$. How exactly am I supposed to do that? I'm thinking there's a trick I'm supposed to use,…
Mane
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Calculating the directional derivative of $ xy^3/(x^3+y^6)$.

$f(x,y)= xy^3/(x^3+y^6)$ if $(x,y)\neq 0$, $f(x,y)=0$ if $(x,y)=0$. Prove that $f'(0; a)$ exists for every vector $a$. I know how to find the directional derivative from limit equation, but don't know how to prove it. Please help.
Ashika
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How to solve when the unknown is given?

A curve has a gradient function $px^2 - 5x$, where $p$ is a constant . The tangent to the curve at the point $x=1$ is parallel to the straight line $y+2x-5=0$. Find the value of $p$.
San San
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Where am I wrong with this derivative?

I want to derivate this function : $$f(t) = \frac{3}{\sin(t)}$$ I know that the derivative of $\frac{u(x)}{v(x)}$is$\frac{u'v-uv'}{v^{2}}$ in general and that in this fraction : $$u'(t) = 0$$ $$v'(t) = \cos(t)$$ So I do : $$\frac{0\times…
Pop Flamingo
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The derivative of $z=x^2+xy+ y^2$

I have got confused about this problem, what I have thought was differentiating this with respect to $x$ gives - $\frac{dz}{dx} = 2x + x \frac{dy}{dx} + y + 2y \frac{dy}{dx}$ But, I came across an online video which did it in this…
M.S.E
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Distributional derivate of $f(t)$

I have the function $f(t)=e^{-|t|}$ And I want to distribution derivate it to $f''(t)$. I am aware of that the $f'(t)$ function will be: But how do I derivate to $f''(t)$ ?
user1223333312
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Show that $f$ is a contraction if and only if there exists $r \in (0, 1)$ such that $|f'(x)| \leq r$, for all $x \in [a, b]$.

Let $f : [a, b] \rightarrow [a, b]$ be differentiable. Show that $f$ is a contraction if and only if there exists $r \in (0, 1)$ such that $|f'(x)| \leq r$, for all $x \in [a, b]$. I managed the "if part", but I really doubt the other way. We…
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