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Chapter 3: Problem 37

Find the derivative of the following functions. $$y=8^{x}$$

Short Answer

Expert verified
Answer: The derivative of the function $$y = 8^x$$ is $$\frac{dy}{dx} = 8^x \cdot \ln(8)$$.

Step by step solution

01

Recall the rule for finding the derivative of an exponential function

For a function of the form $$y = a^x$$, where a is a constant, the derivative is given by: $$\frac{dy}{dx} = a^x \cdot \ln(a)$$ In our case, $$a = 8$$, so the rule for finding the derivative of our function is: $$\frac{dy}{dx} = 8^x \cdot \ln(8)$$
02

Apply the rule to our function

Now, we just need to apply the rule from step 1 to our given function, $$y = 8^x$$. So, we have: $$\frac{dy}{dx} = 8^x \cdot \ln(8)$$ That's it! The derivative of the function $$y = 8^x$$ is $$\frac{dy}{dx} = 8^x \cdot \ln(8)$$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponential Functions
Exponential functions are mathematical functions of the form \(y = a^x\) where \(a\) is a constant known as the base and \(x\) is the variable exponent. These functions are widespread in various fields because they represent growth or decay processes, such as population growth or radioactive decay. One of the key characteristics of exponential functions is their constant growth rate, meaning the rate of change of the function is proportional to the current value of the function itself.

To identify an exponential function, look for a variable exponent with a constant base. For example, in the function \(y = 8^x\), the base is 8 and it remains constant, while \(x\) varies. This creates a swiftly increasing or decreasing curve, depending on whether the base is greater or less than one. Understanding how to differentiate these functions helps in determining tangential slopes, optimizing problems, and analyzing data trends in predictions.

Exponential functions are foundational in calculus, making it crucial for you to understand their properties and behaviors.
Differentiation Rules
Differentiation is a core concept in calculus that allows us to find the rate of change of a function. With functions like \(y = a^x\), differentiation rules provide a systematic way to calculate derivatives efficiently without always reverting to first principles.

For exponential functions, the derivative can be found using a specific rule: for any function \(y = a^x\), the derivative \(\frac{dy}{dx}\) is calculated as \(a^x \cdot \ln(a)\). In this rule:
  • \(a^x\) represents the original function.
  • \(\ln(a)\) is the natural logarithm of the base \(a\).
The derivative gives us the function's rate of change concerning \(x\). For example, applying this rule, the derivative of \(y = 8^x\) is \(8^x \cdot \ln(8)\). This means for any value of \(x\), the slope of the tangent to the function curve at that point can be calculated using this derivative.

Mastering these differentiation rules simplifies solving complex calculus problems and is essential for dealing with exponential growth and decay scenarios.
Natural Logarithm
The natural logarithm, denoted as \(\ln\), is the logarithm to the base \(e\), where \(e\) is approximately 2.71828. This constant \(e\) is unique and crucial in mathematics because it frequently appears in calculations involving growth and decay, compounding, and in differential equations.

In the context of differentiating exponential functions, the natural logarithm plays a significant role. The derivative of an exponential function \(y = a^x\) relies on multiplying the original function \(a^x\) by \(\ln(a)\). This operation comes from the chain rule of differentiation as applied to exponential functions, which essentially tells us how the function's growth rate is scaled by the logarithm of the base.
  • \(\ln(a)\) modifies the exponential rate of change depending on \(a\).
  • Utilizing \(\ln\) makes derivatives of exponential functions manageable.
Understanding the behavior and properties of \(\ln\) helps explain why certain functions grow faster or slower and assists in optimizing processes involving exponential growth, such as finance, biology, or physics.|

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Most popular questions from this chapter

Carry out the following steps. a. Use implicit differentiation to find \(\frac{d y}{d x}\). b. Find the slope of the curve at the given point. $$x \sqrt[3]{y}+y=10 ;(1,8)$$

Let \(f\) and \(g\) be differentiable functions with \(h(x)=f(g(x)) .\) For a given constant \(a,\) let \(u=g(a)\) and \(v=g(x),\) and define $$H(v)=\left\\{\begin{array}{ll} \frac{f(v)-f(u)}{v-u}-f^{\prime}(u) & \text { if } v \neq u \\ 0 & \text { if } v=u \end{array}\right.$$ a. Show that \(\lim _{y \rightarrow u} H(v)=0\) b. For any value of \(u,\) show that $$f(v)-f(u)=\left(H(v)+f^{\prime}(u)\right)(v-u)$$ c. Show that $$h^{\prime}(a)=\lim _{x \rightarrow a}\left(\left(H(g(x))+f^{\prime}(g(a))\right) \cdot \frac{g(x)-g(a)}{x-a}\right)$$ d. Show that \(h^{\prime}(a)=f^{\prime}(g(a)) g^{\prime}(a)\)

Vibrations of a spring Suppose an object of mass \(m\) is attached to the end of a spring hanging from the ceiling. The mass is at its equilibrium position \(y=0\) when the mass hangs at rest. Suppose you push the mass to a position \(y_{0}\) units above its equilibrium position and release it. As the mass oscillates up and down (neglecting any friction in the system , the position y of the mass after t seconds is $$y=y_{0} \cos (t \sqrt{\frac{k}{m}})(4)$$ where \(k>0\) is a constant measuring the stiffness of the spring (the larger the value of \(k\), the stiffer the spring) and \(y\) is positive in the upward direction. A mechanical oscillator (such as a mass on a spring or a pendulum) subject to frictional forces satisfies the equation (called a differential equation) $$y^{\prime \prime}(t)+2 y^{\prime}(t)+5 y(t)=0$$ where \(y\) is the displacement of the oscillator from its equilibrium position. Verify by substitution that the function \(y(t)=e^{-t}(\sin 2 t-2 \cos 2 t)\) satisfies this equation.

The following equations implicitly define one or more functions. a. Find \(\frac{d y}{d x}\) using implicit differentiation. b. Solve the given equation for \(y\) to identify the implicitly defined functions \(y=f_{1}(x), y=f_{2}(x), \ldots.\) c. Use the functions found in part (b) to graph the given equation. \(y^{2}(x+2)=x^{2}(6-x)\) (trisectrix)

Assume \(f\) is a differentiable function whose graph passes through the point \((1,4) .\) Suppose \(g(x)=f\left(x^{2}\right)\) and the line tangent to the graph of \(f\) at (1,4) is \(y=3 x+1 .\) Find each of the following. a. \(g(1)\) b. \(g^{\prime}(x)\) c. \(g^{\prime}(1)\) d. An equation of the line tangent to the graph of \(g\) when \(x=1\)
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