Differentiate The Function $g(x) = 7x^2 + 3 \tan X$.Answer: $\square$

Differentiate the Function $g(x) = 7x^2 + 3 \tan x$

In this article, we will focus on differentiating the given function $g(x) = 7x^2 + 3 \tan x$. Differentiation is a fundamental concept in calculus, and it plays a crucial role in various fields such as physics, engineering, and economics. The derivative of a function represents the rate of change of the function with respect to its input variable.

What is Differentiation?

Differentiation is a mathematical process that helps us find the derivative of a function. The derivative of a function $f(x)$ is denoted by $f'(x)$ and represents the rate of change of the function with respect to its input variable $x$. In other words, the derivative of a function tells us how fast the function changes as its input variable changes.

The Function $g(x) = 7x^2 + 3 \tan x$

The given function is $g(x) = 7x^2 + 3 \tan x$. This function consists of two terms: $7x^2$ and $3 \tan x$. The first term is a quadratic function, while the second term is a trigonometric function.

Differentiating the Quadratic Term

To differentiate the quadratic term $7x^2$, we will use the power rule of differentiation. The power rule states that if $f(x) = x^n$, then $f'(x) = nx^{n-1}$. In this case, $n = 2$, so we have:

$$\frac{d}{dx}(7x^2) = 7 \cdot 2x^{2-1} = 14x$$

Differentiating the Trigonometric Term

To differentiate the trigonometric term $3 \tan x$, we will use the chain rule of differentiation. The chain rule states that if $f(x) = g(h(x))$, then $f'(x) = g'(h(x)) \cdot h'(x)$. In this case, $g(x) = 3x$ and $h(x) = \tan x$. We have:

$$\frac{d}{dx}(3 \tan x) = 3 \cdot \sec^2 x \cdot \frac{d}{dx}(\tan x) = 3 \sec^2 x$$

Differentiating the Function $g(x)$

Now that we have differentiated the quadratic and trigonometric terms, we can differentiate the function $g(x) = 7x^2 + 3 \tan x$ by adding the derivatives of the two terms:

$$\frac{d}{dx}(g(x)) = \frac{d}{dx}(7x^2 + 3 \tan x) = 14x + 3 \sec^2 x$$

In this article, we differentiated the function $g(x) = 7x^2 + 3 \tan x$ using the power rule and the chain rule of differentiation. We found that the derivative of the function is $14x + 3 \sec^2 x$. This result demonstrates the importance of differentiation in calculus and its applications in various fields.

The final answer is $\boxed{14x + 3 \sec^2 x}$.

• [1] Calculus, 3rd edition, Michael Spivak
• [2] Calculus, 2nd edition, James Stewart
• [3] Trigonometry, 2nd edition, Charles P. McKeague

• Differentiation of trigonometric functions
• Differentiation of exponential functions
• Differentiation of logarithmic functions
• Applications of differentiation in physics and engineering

• [1] Calculus, 3rd edition, Michael Spivak
• [2] Calculus, 2nd edition, James Stewart
• [3] Trigonometry, 2nd edition, Charles P. McKeague

• Derivative: The rate of change of a function with respect to its input variable.
• Power rule: A rule of differentiation that states if $f(x) = x^n$, then $f'(x) = nx^{n-1}$.
• Chain rule: A rule of differentiation that states if $f(x) = g(h(x))$, then $f'(x) = g'(h(x)) \cdot h'(x)$.
• Secant: The reciprocal of the cosine function.
  Q&A: Differentiation of the Function $g(x) = 7x^2 + 3 \tan x$

In our previous article, we differentiated the function $g(x) = 7x^2 + 3 \tan x$ using the power rule and the chain rule of differentiation. In this article, we will answer some frequently asked questions related to the differentiation of this function.

Q: What is the derivative of the function $g(x) = 7x^2 + 3 \tan x$?

A: The derivative of the function $g(x) = 7x^2 + 3 \tan x$ is $14x + 3 \sec^2 x$.

Q: How do I differentiate the quadratic term $7x^2$?

A: To differentiate the quadratic term $7x^2$, you can use the power rule of differentiation. The power rule states that if $f(x) = x^n$, then $f'(x) = nx^{n-1}$. In this case, $n = 2$, so we have:

$$\frac{d}{dx}(7x^2) = 7 \cdot 2x^{2-1} = 14x$$

Q: How do I differentiate the trigonometric term $3 \tan x$?

A: To differentiate the trigonometric term $3 \tan x$, you can use the chain rule of differentiation. The chain rule states that if $f(x) = g(h(x))$, then $f'(x) = g'(h(x)) \cdot h'(x)$. In this case, $g(x) = 3x$ and $h(x) = \tan x$. We have:

$$\frac{d}{dx}(3 \tan x) = 3 \cdot \sec^2 x \cdot \frac{d}{dx}(\tan x) = 3 \sec^2 x$$

Q: Can I use the product rule to differentiate the function $g(x) = 7x^2 + 3 \tan x$?

A: Yes, you can use the product rule to differentiate the function $g(x) = 7x^2 + 3 \tan x$. However, in this case, it is more efficient to use the power rule and the chain rule of differentiation.

Q: What is the significance of the derivative of the function $g(x) = 7x^2 + 3 \tan x$?

A: The derivative of the function $g(x) = 7x^2 + 3 \tan x$ represents the rate of change of the function with respect to its input variable. This is a fundamental concept in calculus and has numerous applications in physics, engineering, and economics.

Q: Can I use the derivative of the function $g(x) = 7x^2 + 3 \tan x$ to find the equation of the tangent line to the graph of the function?

A: Yes, you can use the derivative of the function $g(x) = 7x^2 + 3 \tan x$ to find the equation of the tangent line to the graph of the function. The equation of the tangent line is given by:

$$y - g(x_0) = g'(x_0)(x - x_0)$$

where $(x_0, g(x_0))$ is a point on the graph of the function.

In this article, we answered some frequently asked questions related to the differentiation of the function $g(x) = 7x^2 + 3 \tan x$. We hope that this article has been helpful in clarifying any doubts you may have had about the differentiation of this function.

The final answer is $\boxed{14x + 3 \sec^2 x}$.

• [1] Calculus, 3rd edition, Michael Spivak
• [2] Calculus, 2nd edition, James Stewart
• [3] Trigonometry, 2nd edition, Charles P. McKeague

• Differentiation of trigonometric functions
• Differentiation of exponential functions
• Differentiation of logarithmic functions
• Applications of differentiation in physics and engineering

• [1] Calculus, 3rd edition, Michael Spivak
• [2] Calculus, 2nd edition, James Stewart
• [3] Trigonometry, 2nd edition, Charles P. McKeague

• Derivative: The rate of change of a function with respect to its input variable.
• Power rule: A rule of differentiation that states if $f(x) = x^n$, then $f'(x) = nx^{n-1}$.
• Chain rule: A rule of differentiation that states if $f(x) = g(h(x))$, then $f'(x) = g'(h(x)) \cdot h'(x)$.
• Secant: The reciprocal of the cosine function.