If F ( X ) = Tan ⁡ X − 4 Sec ⁡ X F(x)=\frac{\tan X-4}{\sec X} F ( X ) = S E C X T A N X − 4 ​ , Find:1. F ′ ( X F^{\prime}(x F ′ ( X ]2. F ′ ( 3 F^{\prime}(3 F ′ ( 3 ]

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If f(x)=tanx4secxf(x)=\frac{\tan x-4}{\sec x}, find: 1. f(x)f^{\prime}(x) 2. f(3)f^{\prime}(3)

In this article, we will explore the concept of finding the derivative of a given function, specifically f(x)=tanx4secxf(x)=\frac{\tan x-4}{\sec x}. The derivative of a function represents the rate of change of the function with respect to its input variable. In this case, we will use the quotient rule and the chain rule to find the derivative of f(x)f(x).

The quotient rule is a fundamental rule in calculus that allows us to find the derivative of a quotient of two functions. The quotient rule states that if we have a function of the form f(x)=g(x)h(x)f(x)=\frac{g(x)}{h(x)}, then the derivative of f(x)f(x) is given by:

f(x)=h(x)g(x)g(x)h(x)[h(x)]2f^{\prime}(x)=\frac{h(x)g^{\prime}(x)-g(x)h^{\prime}(x)}{[h(x)]^{2}}

To find the derivative of f(x)=tanx4secxf(x)=\frac{\tan x-4}{\sec x}, we can use the quotient rule. Let g(x)=tanx4g(x)=\tan x-4 and h(x)=secxh(x)=\sec x. Then, we have:

g(x)=sec2x and h(x)=secxtanxg^{\prime}(x)=\sec^{2}x \text{ and } h^{\prime}(x)=\sec x \tan x

Using the quotient rule, we get:

f(x)=(secx)(sec2x)(tanx4)(secxtanx)(secx)2f^{\prime}(x)=\frac{(\sec x)(\sec^{2}x)-(\tan x-4)(\sec x \tan x)}{(\sec x)^{2}}

Simplifying the expression, we get:

f(x)=sec3xtan2xsecx+4secxtanx(secx)2f^{\prime}(x)=\frac{\sec^{3}x-\tan^{2}x \sec x+4 \sec x \tan x}{(\sec x)^{2}}

To find the value of f(3)f^{\prime}(3), we need to substitute x=3x=3 into the expression for f(x)f^{\prime}(x). We get:

f(3)=sec3(3)tan2(3)sec(3)+4sec(3)tan(3)(sec(3))2f^{\prime}(3)=\frac{\sec^{3}(3)-\tan^{2}(3) \sec(3)+4 \sec(3) \tan(3)}{(\sec(3))^{2}}

Using a calculator to evaluate the expression, we get:

f(3)=2.18541.73212.1854+42.18540.7265(2.1854)2f^{\prime}(3)=\frac{2.1854-1.7321 \cdot 2.1854+4 \cdot 2.1854 \cdot 0.7265}{(2.1854)^{2}}

Simplifying the expression, we get:

f(3)=2.18543.7943+6.31334.7559f^{\prime}(3)=\frac{2.1854-3.7943+6.3133}{4.7559}

f(3)=4.70444.7559f^{\prime}(3)=\frac{4.7044}{4.7559}

f(3)=0.9873f^{\prime}(3)=0.9873

In this article, we used the quotient rule and the chain rule to find the derivative of the function f(x)=tanx4secxf(x)=\frac{\tan x-4}{\sec x}. We then used the expression for f(x)f^{\prime}(x) to find the value of f(3)f^{\prime}(3). The result shows that the derivative of f(x)f(x) at x=3x=3 is approximately 0.98730.9873.

  • [1] Stewart, J. (2016). Calculus: Early Transcendentals. Cengage Learning.
  • [2] Anton, H. (2017). Calculus: Single Variable. John Wiley & Sons.

The value of f(3)f^{\prime}(3) is an approximation, as the expression for f(x)f^{\prime}(x) involves trigonometric functions that are evaluated at x=3x=3. The actual value of f(3)f^{\prime}(3) may be different depending on the specific values of the trigonometric functions at x=3x=3.
Q&A: If f(x)=tanx4secxf(x)=\frac{\tan x-4}{\sec x}, find: 1. f(x)f^{\prime}(x) 2. f(3)f^{\prime}(3)

In our previous article, we explored the concept of finding the derivative of a given function, specifically f(x)=tanx4secxf(x)=\frac{\tan x-4}{\sec x}. We used the quotient rule and the chain rule to find the derivative of f(x)f(x) and then evaluated the derivative at x=3x=3. In this article, we will answer some common questions related to the derivative of f(x)f(x).

Q: What is the quotient rule?

A: The quotient rule is a fundamental rule in calculus that allows us to find the derivative of a quotient of two functions. The quotient rule states that if we have a function of the form f(x)=g(x)h(x)f(x)=\frac{g(x)}{h(x)}, then the derivative of f(x)f(x) is given by:

f(x)=h(x)g(x)g(x)h(x)[h(x)]2f^{\prime}(x)=\frac{h(x)g^{\prime}(x)-g(x)h^{\prime}(x)}{[h(x)]^{2}}

Q: How do I apply the quotient rule to find the derivative of f(x)f(x)?

A: To apply the quotient rule, we need to identify the functions g(x)g(x) and h(x)h(x) in the given function f(x)=tanx4secxf(x)=\frac{\tan x-4}{\sec x}. In this case, we have g(x)=tanx4g(x)=\tan x-4 and h(x)=secxh(x)=\sec x. We then need to find the derivatives of g(x)g(x) and h(x)h(x), which are g(x)=sec2xg^{\prime}(x)=\sec^{2}x and h(x)=secxtanxh^{\prime}(x)=\sec x \tan x. Finally, we can use the quotient rule to find the derivative of f(x)f(x).

Q: What is the chain rule?

A: The chain rule is a fundamental rule in calculus that allows us to find the derivative of a composite function. The chain rule states that if we have a function of the form f(x)=g(h(x))f(x)=g(h(x)), then the derivative of f(x)f(x) is given by:

f(x)=g(h(x))h(x)f^{\prime}(x)=g^{\prime}(h(x)) \cdot h^{\prime}(x)

Q: How do I apply the chain rule to find the derivative of f(x)f(x)?

A: To apply the chain rule, we need to identify the inner function h(x)h(x) and the outer function g(x)g(x) in the given function f(x)=tanx4secxf(x)=\frac{\tan x-4}{\sec x}. In this case, we have h(x)=secxh(x)=\sec x and g(x)=tanx4secxg(x)=\frac{\tan x-4}{\sec x}. We then need to find the derivatives of h(x)h(x) and g(x)g(x), which are h(x)=secxtanxh^{\prime}(x)=\sec x \tan x and g(x)=sec2xtan2xsecx+4secxtanx(secx)2g^{\prime}(x)=\frac{\sec^{2}x-\tan^{2}x \sec x+4 \sec x \tan x}{(\sec x)^{2}}. Finally, we can use the chain rule to find the derivative of f(x)f(x).

Q: What is the value of f(3)f^{\prime}(3)?

A: To find the value of f(3)f^{\prime}(3), we need to substitute x=3x=3 into the expression for f(x)f^{\prime}(x). We get:

f(3)=sec3(3)tan2(3)sec(3)+4sec(3)tan(3)(sec(3))2f^{\prime}(3)=\frac{\sec^{3}(3)-\tan^{2}(3) \sec(3)+4 \sec(3) \tan(3)}{(\sec(3))^{2}}

Using a calculator to evaluate the expression, we get:

f(3)=2.18541.73212.1854+42.18540.7265(2.1854)2f^{\prime}(3)=\frac{2.1854-1.7321 \cdot 2.1854+4 \cdot 2.1854 \cdot 0.7265}{(2.1854)^{2}}

Simplifying the expression, we get:

f(3)=2.18543.7943+6.31334.7559f^{\prime}(3)=\frac{2.1854-3.7943+6.3133}{4.7559}

f(3)=4.70444.7559f^{\prime}(3)=\frac{4.7044}{4.7559}

f(3)=0.9873f^{\prime}(3)=0.9873

In this article, we answered some common questions related to the derivative of the function f(x)=tanx4secxf(x)=\frac{\tan x-4}{\sec x}. We used the quotient rule and the chain rule to find the derivative of f(x)f(x) and then evaluated the derivative at x=3x=3. The result shows that the derivative of f(x)f(x) at x=3x=3 is approximately 0.98730.9873.

  • [1] Stewart, J. (2016). Calculus: Early Transcendentals. Cengage Learning.
  • [2] Anton, H. (2017). Calculus: Single Variable. John Wiley & Sons.

The value of f(3)f^{\prime}(3) is an approximation, as the expression for f(x)f^{\prime}(x) involves trigonometric functions that are evaluated at x=3x=3. The actual value of f(3)f^{\prime}(3) may be different depending on the specific values of the trigonometric functions at x=3x=3.