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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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finding the derivative of ${{\sqrt x}(x^2 - {\sqrt x})}$

I am trying to find the derivative of this expression ${{\sqrt x}(x^2 - {\sqrt x})}$ I would first of simplify the expression to: ${x^{1\over2}(x^2 - x^{1\over2})}$ And then apply ${x^{1\over2}}$ to the term in the brackets" => ${x^{3 \over2} -…
dagda1
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Derivative of dot product

Let $\frac{dy}{dt}=Ay+g(y)$ and consider $\lVert y(t)\rVert^2=\langle y,y\rangle$. I would like to prove that $$ \frac{d}{dt}\langle y,y\rangle= 2\langle\frac{dy}{dt},y\rangle. $$ To do so, I made the following start: $$ \langle y+\Delta y,y+\Delta…
Salamo
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Use MVT to compare the value of two numbers

Use MVT to Prove that $.99^5>= .95$ I realize I should find a function before using MVT. However, the only function I can think about is $f(x)=x^5$, which doesn't work in this case. Any idea how about how to find a proper function?
XXWANGL
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To find the $n$th derivative of this function.

Let $f(x)$ be smooth and continuous for $|x|<1$. I am interested in the $n$th derivative of: $$g(x) = f(x) e^{af(x)}$$ for some $a>0$. Is it possible to write this in a neat form? Thanks.
user202654
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Related rates: Two planes converging towards a point

An air traffic controller spots two planes at the same altitude converging on a point as they fly at right angles to each other. One plane is 225 miles from the point and is moving at 450 miles per hour. The other plane is 300 miles from the point…
Haim
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f(a) = a using the mean value theorem

Say that $f$ is differentiable and that the derivative of $f$ does not equal $1$ on $(-\infty, \infty)$. Show that there is at most one real number a such that $f(a)=a$. In order to solve this I am required to use the mean value theorem. I…
slehmann
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surface area of a cube related rates problem

The volume of a cube is expanding at a rate of $4cm^3/s$. What is the rate the surface area is changing when the area is $24cm^2$. What I do…
user3258845
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Find all values of $\theta$ such that the tangent line to $f(\theta) = \theta \sin(\theta)$ is given by $y = \theta$

$$f(\theta) = \theta \sin(\theta)$$ Derived to: $$f'(\theta) = \sin(\theta) + \theta \cos(\theta)$$ How do I solve it from here?
user10984587
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Differentiation and second derivative

Let $f$ be a function continuous on $[0, 1]$ and twice differentiable on $(0,1)$. Suppose that: $f(0) = f(1) = 0$ and $f(c) >0$ for some $c \in (0,1)$. Prove that there exists $x_{0}\in(0,1)$ such that $f''(x_{0}) < 0$. I'm not sure how to…
Jason Tan
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Calculate the value of a derivative at the origin

I have the following question in a course: An example of the logistic function is defined by $$\varphi(v)=\frac{1}{1+e^{-av}}$$ whose limiting values are $0$ and $1$. Show that the derivative of $\varphi(v)$ with respect to $v$ is given by…
EFK10003
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A version of the product rule

Using the product rule we know that $$\frac{ {\rm d}\ln(fg)}{ {\rm d} x} = \frac{f'g+fg'}{fg}$$ Is there a function $K$ such that $$\frac{ {\rm d} K(f,g)}{ {\rm d} x} = \frac{f'g-fg'}{fg}$$ ...?
Charles Hagwood
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Chain rule for a function of a multivariable function

I've got confused in something which should not be too confusing. But... If we have some function $f(r)$, where $r=\sqrt{x^2+y^2+z^2}$, then what is $f'(r) = \frac{df}{dr}$? I thought it would be this: $\frac{df}{dr} = \frac{∂r}{∂x} + \frac{∂r}{∂x}…
sequence
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Application of Differentiation

Prove that if the curve $y = x^3 + px + q$ is tangent to the x-axis, then $$4p^3 + 27q^2 = 0$$ I differentiated $y$ and obtained the value $3x^2 + p$. If the curve is tangent to the x-axis, it implies that $x=0$ (or is it $y = 0$?). How do I…
user271716
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Existence of a measurable $\theta$ such that $\frac{f(y)-f(x)}{y-x}=f'(\theta_{x,y})$?

Suppose $f:]a,b[\to\Bbb R$ is differentiable (possibly $C^1$, $C^2$ or a lot smoother, say $C^\infty$), and define $T=\lbrace(x,y)\mid a
Olivier Bégassat
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Check Differentiability

chech whether the function is differentiable at $x=0$ $$f(x)=\left\lbrace \begin{array}{cl} \arctan\frac{1}{\left | x \right |}, & x\neq 0 \\ \frac{\pi}{2}, & x=0\\ \end{array}\right.$$ I feel that this is differentable at given point but I am…
user229886
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