Find the derivative of $y=x \sin ^{-1} x+\sqrt{1-x^2}$. Simplify where possible.

Find the derivative of the function. Simplify where possible.

Differentiate the function.

$$h(x)=e^{x^{2}+\ln x}$$

Differentiate the function.

$$y=\log _{8}\left(x^{2}+3 x\right)$$

Differentiate $y=e^{x+1}+1$.

Consider $x e^y=y-1$. Calculate $y^{\prime}$.

Find the derivative of the function.

$$g(x)=\sin \left(\frac{e^{x}}{1+e^{x}}\right)$$

Find the limit.

$$\lim _ { x \rightarrow 0 } \sec x$$

How do you find the limit of x/sinx as x approaches 0?

Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why. lim x→0 x+sinx / x+cosx

Find the limit or show that it does not exist. lim (x2 + 2x7) x-->-x

Find the limit. lim (1 - cos x)/ sinx x→0

If H(θ) = θ cos θ, find H'(θ) and H''(θ).

Let α be the angle between vectors a and b, where a = (cosθ)i + (sinθ) j and b = (sinθ)i + (cosθ) j, with 0 < θ < π / 4. Express α in terms of θ.