Math Problem Statement

Derivsigve of 2sec squared 5x plus 2x

Solution

To find the derivative of 2sec2(5x)+2x2 \sec^2(5x) + 2x, we will use the following rules:

  1. The derivative of sec2(u)\sec^2(u) with respect to uu is 2sec2(u)tan(u)2\sec^2(u) \tan(u).
  2. The chain rule: ddxf(g(x))=f(g(x))g(x)\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x).
  3. The derivative of 2x2x is simply 22.

Let's differentiate each term:

Step 1: Differentiate 2sec2(5x)2 \sec^2(5x)

First, apply the chain rule. The outer function is sec2(5x)\sec^2(5x) and the inner function is 5x5x.

ddx[2sec2(5x)]=2(2sec2(5x)tan(5x))ddx(5x)\frac{d}{dx} [2 \sec^2(5x)] = 2 \cdot \left(2 \sec^2(5x) \tan(5x)\right) \cdot \frac{d}{dx}(5x)

The derivative of 5x5x is 55, so:

=22sec2(5x)tan(5x)5=20sec2(5x)tan(5x)= 2 \cdot 2 \sec^2(5x) \tan(5x) \cdot 5 = 20 \sec^2(5x) \tan(5x)

Step 2: Differentiate 2x2x

The derivative of 2x2x is simply 22.

Step 3: Combine both results

So the derivative of 2sec2(5x)+2x2 \sec^2(5x) + 2x is:

20sec2(5x)tan(5x)+220 \sec^2(5x) \tan(5x) + 2

Final Answer:

ddx(2sec2(5x)+2x)=20sec2(5x)tan(5x)+2\frac{d}{dx} \left( 2 \sec^2(5x) + 2x \right) = 20 \sec^2(5x) \tan(5x) + 2

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:

  1. How would you differentiate sec(3x2+1)\sec(3x^2 + 1)?
  2. What is the derivative of tan(x2)+x3\tan(x^2) + x^3?
  3. How does the chain rule apply to the function sin2(4x)\sin^2(4x)?
  4. Can you find the second derivative of 2sec2(5x)+2x2 \sec^2(5x) + 2x?
  5. What are some applications of the derivative of sec2(x)\sec^2(x) in physics?

Tip: Always identify inner and outer functions clearly when using the chain rule for derivatives.

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Math Problem Analysis

Mathematical Concepts

Calculus
Differentiation
Chain Rule
Trigonometric Functions

Formulas

Derivative of sec^2(u): 2 * sec^2(u) * tan(u)
Chain Rule: d/dx[f(g(x))] = f'(g(x)) * g'(x)
Derivative of a linear function: d/dx[2x] = 2

Theorems

Chain Rule
Derivative of Trigonometric Functions

Suitable Grade Level

Grades 11-12 / Early College