Actually,
you did not upload an image with a task — you uploaded a reference sheet titled “DIFFERENTIATION RULES” that lists 25 standard derivative formulas.
Since there is no actual *problem* or *task* to solve (no function to differentiate, no equation to analyze, no question asked), I cannot “solve the problem” as requested.
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✔ What I can do instead:
I’ll explain what this sheet is for and how to use it — which may be what you’re looking for!
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## 📘 What This Sheet Is
This is a
cheat sheet of differentiation rules, commonly used in calculus courses. It organizes the most important derivative formulas into categories:
1.
Basic Rules — constants, constant multiples, power rule, sum/difference, product, quotient, chain rule.
2.
Trigonometric Functions — derivatives of sin, cos, tan, cot, sec, csc.
3.
Inverse Trigonometric Functions — derivatives of arcsin, arccos, arctan, etc.
4.
Exponential and Logarithmic Functions — derivatives of ln, logₐ, eᵘ, aᵘ, √u, |u|.
Each rule is written in
Leibniz notation (d/dx) and often includes the chain rule version (e.g., du/dx).
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## 🔍 How to Use This Sheet
Suppose you are asked to find the derivative of:
>
f(x) = 3x⁴ + sin(2x)
You would:
- Use
Rule 4 (Sum Rule) → d/dx [3x⁴] + d/dx [sin(2x)]
- Use
Rule 2 (Constant Multiple) +
Rule 3 (Power Rule) → 3 × 4x³ = 12x³
- Use
Rule 8 (Derivative of sin u) + Chain Rule → cos(2x) × d/dx(2x) = cos(2x) × 2
- Final answer:
f’(x) = 12x³ + 2cos(2x)
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## ❗ Important Notes
-
Rule 7 is the
Chain Rule: d/dx [f(g(x))] = f’(g(x)) · g’(x)
-
Rules 14–19 have absolute values or square roots in denominators — pay attention to domain restrictions!
-
Rule 25 for |u|: The derivative is u/|u| × du/dx, which equals sign(u) × du/dx (except at u=0, where it’s undefined).
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##
✔ If You Meant to Ask Something Else…
If you intended to upload a specific problem (like “Differentiate y = x² ln x” or “Find dy/dx for y = tan⁻¹(3x)”), please:
➡️ Upload the image again (or type the problem here)
➡️ I’ll solve it step-by-step using this very sheet!
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Let me know how else I can help — happy to walk through any example! 😊
Parent Tip: Review the logic above to help your child master the concept of ap calculus derivative worksheet.
