∫ calculus

1. Stating the problem: Find the derivative of the function $$y = x^2 \sin^4 x + x \cos^{-2} x.$$\n\n2. Rewrite the function for clarity: $$y = x^2 (\sin x)^4 + x (\cos x)^{-2}.$$\

1. We are given the function $$r=6(\sec \theta -\tan \theta)^{3/2}$$ and asked to find its derivative with respect to $$\theta$$.\n\n2. Let $$u=\sec \theta - \tan \theta$$, then $$

1. **حساب النهايات:** أ) $$A=\lim_{x\to+\infty}\frac{x}{\sqrt[3]{x^2+1}}$$

1. Let's analyze the problem: We have a smooth continuous function graphed on the xy-plane from $x=-1$ to $x=8$ and $y=-1$ to $y=8$. 2. The curve's behavior is described as startin

1. The problem asks us to find the derivative of the function $$r = (\csc \theta + \cot \theta)^{-1}$$ with respect to $$\theta$$. 2. Recall the derivative rules: the derivative of

1. The problem is to find where the curve is concave up, concave down, and identify the inflection points. 2. Concavity depends on the second derivative $f''(x)$ of the function re

1. **Problem statement:** (a) Find the fifth Taylor polynomial for $f(x) = \ln(x+1)$ centered at $x=0$.

1. **State the problem:** We need to determine where the function $f$ is increasing and decreasing, identify the concavity intervals (where it is concave up or down), and find the

1. The problem is to find the derivative of the function $$q = (2r - r^2)^{1/3}$$ with respect to $$r$$. 2. Let $$u = 2r - r^2$$, so the function becomes $$q = u^{1/3}$$.

1. The problem asks for the derivative of the function given in question 20. 2. Since the user did not provide the function from question 20, please supply the function or state th

1. نبدأ بكتابة المعادلة المعطاة: $$ (ن (س) + (س + س)^2) dy = س^3 - 7س + ج $$ 2. نلاحظ أن السؤال يطلب قيمة $ن^{(3)}$ أي ربما يعني المشتقة الثالثة لدالة ن بالنسبة لـ $س$.

1. **Statement of the problem:** We are given the function $$d(s) = \cos(\cos s)$$ for $$s \in [0,7\pi]$$.

1. نقرأ كما هو مطلوب في السؤال: إذا كانت د(س) دالة والمشتقة د (س) تعطينا د (5) = 0 و د (1) = 5. المطلوب إيجاد \( أذن(5) \). 2. قد يكون في السؤال خطأ بسيط أو لبس في الرموز، ولكن عاد

1. The problem asks us to find the limit \( \lim_{n \to +\infty} \frac{\ln(n)}{\sqrt{n+1}} \). 2. As \( n \to +\infty \), both \( \ln(n) \) and \( \sqrt{n+1} \) grow without bound,

1. المعطى يقول إننا قربنا الدالة ن(س) بالقصران ن(س^3). 2. لدينا ن(٥) = ه، ون(١) = ٠.

1. نبدأ بتحديد المعطيات: دالة ن(س) تقريباً قابلة للاشتقاق على مجالها، ونُعلم أن نقطة (-3, 4) تُمَيّز منحناها، أي أن ن(-3) = 4. 2. المعلومة الثانية هي أن قيمة المشتقة عند النقطة 2 ت

1. Problem: Compute $$\frac{f(x+h)-f(x)}{h}$$ at $$h=0$$ (finding the derivative as a limit) for the function $$f(x) = 3x^2 + 5$$. Step 1: Find $$f(x+h) = 3(x+h)^2 + 5 = 3(x^2 + 2x

1. **State the problem:** Find the derivative of the function $$f(x)=(4+x)^{6+x}, x>-4.$$ We need to express $$f'(x)$$ in terms of $$x$$.

1. Stating the problem: Find the derivative of the function $$f(x)=e^{3x} \sin(\ln(9+x))$$. 2. Use the product rule for differentiation: if $$f(x)=u(x)v(x)$$, then $$f'(x)=u'(x)v(x

1. **State the problem:** We need to find the derivative of the function $$f(x)=e^{\sin^3(7x)}$$. 2. **Rewrite the function:** The function can be written as $$f(x)=e^{(\sin(7x))^3

1. **State the problem:** We need to find the derivative of the function $$f(x) = \frac{e^x}{x^3 + 2 \sin(x)}$$. 2. **Identify components:** This is a quotient of two functions: