Differentiate $$y=\cos\left(\sqrt{2*x^2+3}\right)$$ with respect to x.

Product and Chain Rule Differentiation

For the circle defined by $$x^2+y^2=25$$, first use implicit differentiation to find $$\frac{dy}{dx}$$, and then determine the second derivative $$\frac{d^2y}{dx^2}$$ at the point (3,4).

Which of the following is an example of a function that, when differentiated using inverse trigonometric differentiation, yields $$\frac{1}{\sqrt{1-x^2}}$$ as its derivative, as illustrated by the provided graph?

I. Expressing $$f(x)=\sqrt{3*x+2}$$ as $$(3*x+2)^{1/2}$$ allows the use of the chain rule for differentiation. II. The derivative is given by $$\frac{1}{2}(3*x+2)^{-1/2}*3$$. III. Simplifying the derivative results in $$\frac{3}{2\sqrt{3*x+2}}$$. Which of the above statements are true?

Average Rate of Change on an Interval

Which of the following is an example of using a secant line to approximate the instantaneous rate of change on a curve?

Using the graph of $$f(x)=\sin(x)$$, find the derivative $$f'(x)$$ using the limit definition of the derivative.

Derivative of a Trigonometric Function

Continuity and Differentiability at a Point

Instantaneous Velocity from Position Function

Derivative of the Sine Function

Rate of Change of a Constant Function

Limit Definition of the Derivative

Refer to the graph of the function $$f(x)=x^2$$ shown above. Compute the average rate of change of f on the interval from $$x=1$$ to $$x=3$$.

Exponential Limit at Negative Infinity

I. $$\frac{(x+3)(x+2)}{(x+3)(x-3)}$$ simplifies to $$\frac{x+2}{x-3}$$ as the common factor $$x+3$$ cancels for $$x\neq -3$$. II. After cancellation, substituting $$x = -3$$ gives $$\frac{-3+2}{-3-3}=\frac{-1}{-6}=\frac{1}{6}$$. III. The function becomes continuous at $$x=-3$$ after cancellation, hence the function’s value and its limit at $$x=-3$$ coincide. Determine which of the above statements provide a correct explanation for finding the limit of the function $$\frac{(x+3)(x+2)}{(x+3)(x-3)}$$ as $$x\to -3$$ using algebraic manipulation.

Limit of a Function with Absolute Value

Evaluating a Limit at a Removable Discontinuity

Properties of Continuous Functions