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Chapter 3: Problem 5

How do you find the derivative of a constant multiplied by a function?

Short Answer

Expert verified
Answer: The derivative of a constant multiplied by a function, with respect to x, is equal to the constant multiplied by the derivative of the function with respect to x: d(a*f(x))/dx = a*d(f(x))/dx

Step by step solution

01

Identify the constant and the function

Let the constant be represented by letter 'a' and the function be 'f(x)'. So, we want to find the derivative of a*f(x) with respect to x.
02

Apply the Constant Rule

According to the constant rule, the derivative of a constant multiplied by a function is equal to the constant multiplied by the derivative of the function. So, the derivative of a*f(x) with respect to x is: d(a*f(x))/dx = a*d(f(x))/dx
03

Final Answer

The derivative of the constant 'a' multiplied by the function 'f(x)', with respect to x, is equal to the constant 'a' multiplied by the derivative of the function 'f(x)' with respect to x: d(a*f(x))/dx = a*d(f(x))/dx

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Most popular questions from this chapter

The following limits equal the derivative of a function \(f\) at a point a. a. Find one possible \(f\) and \(a\) b. Evaluate the limit. $$\lim _{h \rightarrow 0} \frac{\cos \left(\frac{\pi}{6}+h\right)-\frac{\sqrt{3}}{2}}{h}$$

Product Rule for the second derivative Assuming the first and second derivatives of \(f\) and \(g\) exist at \(x\), find a formula for \(\frac{d^{2}}{d x^{2}}(f(x) g(x))\)

Let $$g(x)=\left\\{\begin{array}{cl} \frac{1-\cos x}{2 x} & \text { if } x \neq 0 \\ a & \text { if } x=0 \end{array}\right.$$ For what values of \(a\) is \(g\) continuous?

Suppose \(f(2)=2\) and \(f^{\prime}(2)=3 .\) Let $$g(x)=x^{2} \cdot f(x) \text { and } h(x)=\frac{f(x)}{x-3}$$ a. Find an equation of the line tangent to \(y=g(x)\) at \(x=2\) b. Find an equation of the line tangent to \(y=h(x)\) at \(x=2\)

Product Rule for three functions Assume that \(f, g,\) and \(h\) are differentiable at \(x\) a. Use the Product Rule (twice) to find a formula for \(\frac{d}{d x}(f(x) g(x) h(x))\) b. Use the formula in (a) to find \(\frac{d}{d x}\left(e^{2 x}(x-1)(x+3)\right)\)
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