Question

trig limits
lim_{\theta\to0} \frac{\sin\theta}{\theta}=1;\lim_{\theta\to0} \frac{\cos\theta - 1}{\theta}=0
linearization of f at a
l(x)=f(a)+f(a)(x - a)
question 7
evaluate: \frac{d}{dx}x^{4}-5x^{3}-35x + 2
4x^{3}-15x^{2}-35
5x^{5}-15x^{4}-35x^{2}+2x
x^{4}-5x^{3}-35x + 2
no correct answer choice is given.
(4x^{3}-15x^{2})(35x + 2)+(x^{4}-5x^{3})(35x + 2)

Explanation:

Step1: Apply power - rule of differentiation

The power - rule states that if $y = x^n$, then $\frac{dy}{dx}=nx^{n - 1}$. For the function $y=x^{4}-5x^{3}-35x + 2$, we differentiate each term separately.
For the term $x^{4}$, using the power - rule, $\frac{d}{dx}(x^{4})=4x^{4 - 1}=4x^{3}$.

Step2: Differentiate the second term

For the term $-5x^{3}$, $\frac{d}{dx}(-5x^{3})=-5\times3x^{3 - 1}=-15x^{2}$.

Step3: Differentiate the third term

For the term $-35x$, $\frac{d}{dx}(-35x)=-35\times1x^{1 - 1}=-35$.

Step4: Differentiate the constant term

For the constant term $2$, $\frac{d}{dx}(2) = 0$ since the derivative of a constant is 0.

Step5: Combine the derivatives of all terms

$\frac{d}{dx}(x^{4}-5x^{3}-35x + 2)=4x^{3}-15x^{2}-35$.

Answer:

A. $4x^{3}-15x^{2}-35$