{"id":12933,"date":"2023-01-02T19:00:45","date_gmt":"2023-01-02T16:00:45","guid":{"rendered":"https:\/\/starlanguageblog.com\/?p=12933"},"modified":"2023-01-02T19:00:45","modified_gmt":"2023-01-02T16:00:45","slug":"derivative-of-3x-explained","status":"publish","type":"post","link":"https:\/\/www.starlanguageblog.com\/derivative-of-3x-explained\/","title":{"rendered":"Derivative of 3^x Explained"},"content":{"rendered":"

Derivative of 3^x Explained<\/h1>\n

The derivative of 3^x is 3^x * ln(3). To see this, you can use the power rule for derivatives: if f(x) = x^n, then f'(x) = n * x^(n-1).<\/p>\n

So, if you let f(x) = 3^x, then f'(x) = x * 3^(x-1). But 3^(x-1) is just another way of writing 3^x, so f'(x) = x * 3^x. And since ln(3) is just the exponent to which you must raise 3 to get 3^x, we can rewrite this as 3^x * ln(3).<\/p>\n

The power rule for derivatives states that if f(x) = x^n, then the derivative of f(x), denoted as f'(x), is given by f'(x) = n * x^(n-1). This is a general formula that you can use to find the derivative of any function of the form f(x) = x^n, where n is a constant.<\/p>\n

For example, suppose you want to find the derivative of f(x) = x^2. Using the power rule, we have:<\/p>\n

f'(x) = 2 * x^(2-1) = 2 * x^1 = 2x<\/p>\n

So the derivative of f(x) = x^2 is f'(x) = 2x.<\/p>\n

Now, let’s consider the function g(x) = 3^x. We can use the power rule to find the derivative of this function as follows:<\/p>\n

g'(x) = x * 3^(x-1)<\/p>\n

But 3^(x-1) is just another way of writing 3^x, so we can rewrite this as:<\/p>\n

g'(x) = x * 3^x<\/p>\n

Finally, since ln(3) is the exponent to which you must raise 3 to get 3^x (that is, 3^ln(3) = 3^x), we can rewrite g'(x) as:<\/p>\n

g'(x) = 3^x * ln(3)<\/p>\n

So the derivative of g(x) = 3^x is g'(x) = 3^x * ln(3).<\/p>\n

How do you differentiate 3^x with respect to x?<\/h2>\n

To differentiate 3^x with respect to x, you can use the power rule for derivatives<\/a>. The power rule states that if f(x) = x^n, then the derivative of f(x), denoted as f'(x), is given by f'(x) = n * x^(n-1).<\/p>\n

Applying the power rule to the function f(x) = 3^x, we have:<\/p>\n

f'(x) = x * 3^(x-1)<\/p>\n

But 3^(x-1) is just another way of writing 3^x, so we can rewrite this as:<\/p>\n

f'(x) = x * 3^x<\/p>\n

Finally, since ln(3) is the exponent to which you must raise 3 to get 3^x (that is, 3^ln(3) = 3^x), we can rewrite f'(x) as:<\/p>\n

f'(x) = 3^x * ln(3)<\/p>\n

So the derivative of f(x) = 3^x with respect to x is f'(x) = 3^x * ln(3).<\/p>\n

Derivative of 3\/x using limit definition<\/h2>\n

To find the derivative of the function f(x) = 3\/x using the limit definition, we can use the following formula:<\/p>\n

f'(x) = lim(h->0) [(3\/(x+h)) – (3\/x)] \/ h<\/p>\n

This formula states that the derivative of f(x) at a point x is equal to the limit of the difference quotient of f(x) as h approaches 0.<\/p>\n

Substituting 3\/x for f(x) and simplifying, we get:<\/p>\n

f'(x) = lim(h->0) [(3\/(x+h)) – (3\/x)] \/ h = lim(h->0) (3\/x – 3\/(x+h)) \/ h = lim(h->0) (3x – 3(x+h)) \/ (xh(x+h)) = lim(h->0) (3x – 3x – 3h) \/ (xh(x+h)) = lim(h->0) (-3h) \/ (xh(x+h)) = lim(h->0) -3 \/ (x(x+h)) = -3 \/ x^2<\/p>\n

Therefore, the derivative of f(x) = 3\/x is f'(x) = -3\/x^2.<\/p>\n","protected":false},"excerpt":{"rendered":"

Derivative of 3^x Explained The derivative of 3^x is 3^x * ln(3). To see this, you can use the power rule for derivatives: if f(x) = x^n, then f'(x) = n * x^(n-1). So, if you let f(x) = 3^x, then f'(x) = x * 3^(x-1). But 3^(x-1) is just another way of writing 3^x, […]<\/p>\n","protected":false},"author":1,"featured_media":12934,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2078],"tags":[2094,2095],"class_list":["post-12933","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-maths","tag-derivative-of-3x","tag-derivative-of-3x-explained"],"yoast_head":"\nDerivative of 3^x Explained - Star Language Blog<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.starlanguageblog.com\/derivative-of-3x-explained\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Derivative of 3^x Explained - Star Language Blog\" \/>\n<meta property=\"og:description\" content=\"Derivative of 3^x Explained The derivative of 3^x is 3^x * ln(3). To see this, you can use the power rule for derivatives: if f(x) = x^n, then f'(x) = n * x^(n-1). So, if you let f(x) = 3^x, then f'(x) = x * 3^(x-1). 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