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.

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The $k$th derivative of $\sin^n x$ as $n \ \sin^{n - k} x$ times a polynomial in $\cos x$

It seems the $k$th derivative of the function $x \mapsto \sin^n x$ can be expressed as $n \ \sin^{n - k} x$ times a polynomial with only even or only odd powers of $\cos x$ (depending on the parity of $k$), with polynomials in $n$ as coefficients,…
Adhemar
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Derivative of a delta function of a function

I am just wondering why this is correct, $$ \frac{d}{dR} \delta[f(t')] = -\frac{1}{c}\frac{d}{df}\delta[f(t')], $$ where $t' = t - \frac{R}{c}$. Is this simply due to chain rule? Or something like $$ \frac{d}{dR} =…
Shinobii
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Problem with this $\frac{d}{dx}(y^3)$

How do you differentiate this equation with respect to $x$? $$x^2=xy^3+2$$ $$\frac{d}{dx}(x^2)=\frac{d}{dx}(xy^3)+\frac{d}{dx}(2)$$ $$2x=x\frac{d}{dx}(y^3)+y^3\frac{d}{dx}(x)+0$$ $$2x=x\frac{d}{dx}(y^3)+y^3$$ Here is the problem I am facing with,…
user550260
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proof of $d(xy) = x(dy) + y(dx)$

I was trying to prove $$d(xy) = x(dy) + y(dx)$$ earlier this morning and I used this post to help me understand the task. I understood the entirety of the post for my calculus class, apart from one step. Considering an area of a rectangle with…
vik1245
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Differentiability of a function at a point

My high-school calculus teacher has asserted that a function $f(x)$ can only fail to be differentiable at a point $x=a$ if one of the following is true: The function is discontinuous at $x=a$: $\lim_{x\to a}f(x) \ne f(a)$ The function has a cusp or…
AJMansfield
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How do you differentiate with respect to y?

Find the gradient of $$z=x^y$$ I understand how to get it with respect to $x$ since $y$ is treated as a constant. But when trying to solve it with respect to $y$, why is it incorrect to implicitly differentiate and use the product…
Random Student
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Finding the derivative of $y= (\ln x)^2$

In my math class, we are beginning to find derivatives of more complex functions. I’ve been trying questions from my textbook as practice. Here are two of them that I’m trying out: $y=(\ln x)^2$. First, we take the power rule. This would make it…
Ella
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Partial derivative of the likelihood function respect to $\sigma^2$

I am having problem doing the partial derivative of the likelihood function which is $L(\mu,\sigma^2)=\frac{1}{\sigma\sqrt{2\pi}^n}\times \exp{(-\frac{1}{2\sigma^2}\sum(x_i-\mu)^2)}$ If the first part has solved that the $\hat{\mu}$ is…
Chen
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gradient of max function

If I have a function $f=\max \{0, y-t\}$, and I want to find the gradient of with respect to $[y \ \ t]$, would that simply be $$ \nabla f = \begin{bmatrix} \max\{0,0\} \\ \max\{1,-1\} \end{bmatrix} = \begin{bmatrix} 0 \\ 1 \end{bmatrix} $$
24n8
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Find the radius of a cylinder of given volume V if its surface area is a minimum.

this question is driving me crazy as I'm not sure how they've got the answer. The surface area is given as $S = 2\pi r^2 + \frac {1}{50r} $ and they are asking for the value of r for which S is minimum. The derivative of this (I hope!) is $4\pi r -…
Lorcan
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Differentiation of $(x,y)\mapsto g(x+h(x,y))$

Let $g:\mathbb{R}\to\mathbb{R}$ and $h:\mathbb{R}^2\to\mathbb{R}$ be two differentiable functions. I would like to compute the differential of $T:\mathbb{R}^2\to\mathbb{R}$ such that $T(x,y)=g(x+h(x,y))$. I get $D T(x,y) = Dg(x+h(x,y))\circ…
grear
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Conclusions about derivability of a function $f(x)=x\left|{\log{x}}\right|$

I want to study the derivability of this function $$f(x)=x\left|{\log{x}}\right|$$ My textbook says the function is defined for $x>0$ (easy to understand for me, the argument of the logarithm must be positive) and it says: "it can certainly be…
Cesare
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Differentiate $e^{7x^3-\frac{5}{3}}$

For this equation I'm using the following property $$f(x)=e^{kx}$$ $$f'(x)=ke^{kx}$$ As well as the product rule $$f(x)=uv$$ $$f'(x)=u'v+uv'$$ I factorize $x$ on $e$'s exponent and then use the first property to…
Pablo
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Proof of dy=f’(x)dx

I’ve been wondering about the usage of $dy=f’(x)dx$ in my textbook. There’s not a single justification of how it is proved and it just states that it is true. Since $dy/dx$ can’t be assumed as a fraction, I’m guessing there’s more to it than just…
Jwnle
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Finding the $n$-th derivative of $a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$

Let $f(x)= a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$. I am trying to find $f^n(x)$. By applying the power rule $n$ times, I get this $$f^n(x)=a_{n}(n\cdot n)x^{n-n}+\cdots+ a_1$$ which I think can be simplified to $$f^n(x)=a_{n}n^2+\cdots+…
Cedric Martens
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