What is Derivative calculus: Definition and 12 Discussions

In mathematics, 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). Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances.
The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. For this reason, the derivative is often described as the "instantaneous rate of change", the ratio of the instantaneous change in the dependent variable to that of the independent variable.
Derivatives can 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.
The process of finding a derivative is called differentiation. The reverse process is called antidifferentiation. The fundamental theorem of calculus relates antidifferentiation with integration. Differentiation and integration constitute the two fundamental operations in single-variable calculus.

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1. Dx/x of quotient by def of derivative

$f(x)=\dfrac{x^2-1}{2x-3}$ ok I just don't see any preview so don't want to add more...
2. Derivative of the square root of the function f(x squared)

I started out by rewriting the function as (f(x^2))^(1/2). I then did chain rule to get 1/2(f(x^2))^(-1/2) *(f'(x^2). - I think I need to go further because it is an x^2 in the function, but not sure.
3. Determine for which x the derivative exists for $$f(x)=arcsin(\sqrt x)$$

Hi there. I have the following function: $$f(x)=arcsin(\sqrt x)$$ I've caculated the derivative to: $$f'(x)=\frac{1}{2\sqrt x\sqrt{ (1-x}}$$ And the domain of f(x) to: $$[0, 1]$$ And the domain of f'(x) to: $$(0, 1)$$ I want to determine for which x the derivative exists but I'm not...
4. A Liouville's theorem and time evolution of ensemble average

With the Liouville's theorem $$\frac{{d\rho }}{{dt}} = \frac{{\partial \rho }}{{\partial t}} + \sum\limits_{a = 1}^{3N} {(\frac{{\partial \rho }}{{\partial {p_a}}}\frac{{d{p_a}}}{{dt}} + \frac{{\partial \rho }}{{\partial {q_a}}}\frac{{d{q_a}}}{{dt}})} = 0$$ when we calculate the time evolution...
5. Implicit differentiation problem

Homework Statement If ##x\sqrt{1+y} + y\sqrt{1+x } = 0##, then prove that ##\frac {dy} {dx} = \frac {-1}{(x-1)^2}##. 2.Relevant Equations: $$\frac {dy} {dx} = - \frac {\left (\frac {\partial f}{\partial x} \right)} {\left( \frac {\partial f} {\partial y} \right)}.$$ 3...
6. Pick a,b,c,d for y=ax^3+bx^2+cx+d that models path of plane.

Homework Statement A plane starts its descent from height ##y =h## at ##x = -L## to land at ##(0,0)##. Choose ##a, b, c, d## so its landing path ##y =ax^3 + bx^2 + cx + d## is "smooth". With ##\frac{\mathrm {d}x}{\mathrm {d}t} = V =##constant, find ##\frac{\mathrm {d}y}{\mathrm {d}t}## and...
7. G

How can I calculate the derivative of this function?

Homework Statement Let f(x) be the function whose graph is shown below (I'll upload the image) Determine f'(a) for a = 1,2,4,7. f'(1) = f'(2) = f'(4) = f'(7) = Use one decimal. Homework Equations f(x+h)-f(x)/h The Attempt at a Solution Hi everybody I was trying to do this function...
8. A

B Quick question about calculus (derivatives)

I thought Differentiation is all about understanding it in a graph. Every time I solve a question on differentiation I visualise it as a graph so it's more logical. After all, that IS what the whole topic is about, right? Or am I just wrong? But when you look at these questions...
9. I Differentiation under the integral in retarded potentials

Hello, friends! I know, thanks to @Hawkeye18 who proved this identity to me, that, if ##\phi:V\to\mathbb{R}## is a bounded measurable function defined on the bounded measurable domain ##V\subset\mathbb{R}^3##, then, for any ##k\in\{1,2,3\}##, \frac{\partial}{\partial r_k}\int_V...
10. Master equation -> diffusion equation

Homework Statement I am trying to understand the derivation of the diffusion equation from the Master equation for a 1D chain. We have an endless 1D discrete chain. State from ##n## can jump to ##n+1## and ##n-1## with equal probabilities. The distance between chain links is ##a##. Homework...
11. How to solve this partial derivative which includes a summation?

I was reading a research paper, and I got stuck at this partial differentiation. Please check the image which I have uploaded. Now, I got stuck at Equation (13). How partial derivative was done, where does summation gone? Is it ok to do derivative wrt Pi where summation also includes Pi...
12. I What Is the Formula for the Position of a Mass Falling Towards a Planet?

Question: Finding the closed formula s(t) that gives the approaching position of an inertial mass to a planet Supposing the mass initially stationary, and far enough and for long enough that it is NOT possible to consider the gravity as constant while it moves closer and closer. Said in a...