Skip to main content
Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 3 - DerivativesProblem 84a
Chapter 3, Problem 84a

Use the given graphs of f and g to find each derivative. <IMAGE>
d/dx (5f(x)+3g(x)) |x=1

Verified step by step guidance
1
Step 1: Understand the problem. We need to find the derivative of the function h(x) = 5f(x) + 3g(x) at x = 1, where f and g are functions whose graphs are provided.
Step 2: Apply the linearity of differentiation. The derivative of a sum of functions is the sum of their derivatives. Therefore, d/dx [5f(x) + 3g(x)] = 5 * d/dx [f(x)] + 3 * d/dx [g(x)].
Step 3: Evaluate the derivatives of f and g at x = 1. From the graphs, determine the slopes of the tangent lines to f(x) and g(x) at x = 1, which represent f'(1) and g'(1) respectively.
Step 4: Substitute the values of f'(1) and g'(1) into the expression from Step 2. This gives us 5 * f'(1) + 3 * g'(1).
Step 5: Calculate the final expression using the values obtained from the graphs. This will give the value of the derivative of h(x) at x = 1.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
5m
Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Derivative

The derivative of a function measures how the function's output value changes as its input value changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. In practical terms, the derivative provides the slope of the tangent line to the graph of the function at a given point.
Recommended video:
05:44
Derivatives

Sum Rule of Derivatives

The sum rule states that the derivative of the sum of two functions is equal to the sum of their derivatives. Mathematically, if f(x) and g(x) are differentiable functions, then d/dx [f(x) + g(x)] = f'(x) + g'(x). This rule simplifies the process of finding derivatives when dealing with expressions that involve the addition of multiple functions.
Recommended video:
05:44
Algebra Rules for Finite Sums

Constant Multiple Rule

The constant multiple rule states that the derivative of a constant multiplied by a function is equal to the constant multiplied by the derivative of the function. Formally, if c is a constant and f(x) is a differentiable function, then d/dx [c * f(x)] = c * f'(x). This rule is essential for differentiating expressions where functions are scaled by constants.
Recommended video:
04:02
The Power Rule
Related Practice
Textbook Question

The following limits represent f'(a) for some function f and some real number a.

Find a possible function f and number a.

lim x🠂0 e^x-1 / x

1
views
Textbook Question

The following limits represent f'(a) for some function f and some real number a.

b. Evaluate the limit by computing f'(a).

lim x🠂0 e^x-1 / x

2
views
Textbook Question

The following limits represent f'(a) for some function f and some real number a.

b. Evaluate the limit by computing f'(a).

lim x🠂1 x¹⁰⁰-1 / x-1

2
views
Textbook Question

{Use of Tech} Cell population The population of a culture of cells after t days is approximated by the function P(t)=1600 / 1 + 7e^−0.02t, for t≥0.

e. Graph the growth rate. When is it a maximum and what is the population at the time that the growth rate is a maximum? 

Textbook Question

Use the given graphs of f and g to find each derivative. <IMAGE>

d/dx (g(f(x))) |x=1

Textbook Question

Use the given graphs of f and g to find each derivative. <IMAGE>

d/dx (f(f(x))) |x=4